Accidents
Only effects aircraft. The chance of the plane being destroyed when landing/taking off.
Fix: Increase reliability of the aircraft.

Air Attack
Damages enemy air, reduces the effects of enemy air superiority. Definitely worth it if you are expecting the enemy to beat your airforce. Much cheaper than building a proper airforce, but not nearly as strong.
How to improve: Adding anti-air to your divisions.

Air Superiority
Essentially, high air superiority advantage gives you, or the enemy, a percentage penalty to both defense and breakthrough. Even without CAS or bombers, the effect of fighters on combat is pretty substantial. Every 50 points of Air superiority advantage = 1% penalty to both. Each fighter = 1 point of Air Superiority. Going down the Superior Firepower -> Airland Battle, increases this penalty by 20%. Yes, a ton of numbers, but to cut it short, if you completely ignore the battle of the skies and your enemy dumps 1000 fighters in the sky, your defense and breakthrough is dropped by 20%. It's not something that can be ignored.

Armor
Armor HALVES soft and hard attack taken if higher than the enemies piercing and increases the amount of damage a division does in combat fairly significantly. However, having one extra armor over the enemies piercing is just as beneficial as having a hundred armor over the enemies piercing so there is no benefit to overstacking the stat.
How to improve: Tanks, mechanized.

Attrition
The higher the attrition, the more equipment is lost. Only effects land units.
Causes: Poor weather and terrain. High resistance levels. Low supply. Small amount due to training.
Fix: Avoid fighting the USSR in the winter.
Have spies lower resistance around the front.
Equip divisions with logistic companies to lower supply usage.
Improve infrastructure around the front, improve ports if you are overseas, improve infrastructure from your capital.
Attach maintenance companies to motorized/tank divisions and increase the reliability of your tanks.

Autonomy Together for Victory ONLY
Every puppet has its own autonomy score. If they gain enough of it, they can decrease their puppet level (basically makes them a far less effective puppet, they give you less stuff) and eventually free themselves. You cannot prevent them from gaining autonomy, only take actions to reduce it by building in their territory or lendleasing them stuff. Keep in mind that under certain events puppets can still free themselves from you. Most notably Britain which will lose the majority of the commonwealth if it switches ideologies. There are ways to prevent it, but don't bother. Takes too long to justify.

Breakthrough
See 'defense' but this works for attacking instead. Higher breakthrough means the division can attack for longer.
How to improve? Tanks, mechanized. Better inf. equipment gives it too, but not nearly enough. This is why attacking with predominantly inf. divisions isn't efficient. If you find yourself tearing your hair out at how slow pushes are going, replace your poor USSR soldier with a tank.

Civil War
Civil war occurs either through pre-scripted events, after a coup or after a certain ideology has gained too much support. The only way to prevent scripted ones is to do the relevant focuses. The only way to prevent a coup or natural civil war is to hire your own ministers, but the AI will never use coups against you. You can also cause a civil war through the decisions. Why? It can be significantly quicker than waiting for 50% popularity for your chosen ideology. Keep in mind that the strength of the civil war is equal to their ideologies popularity. If you only have 10% fascist popularity, your civil war will only have 10% of your original military.

Close Air Support
CAS can directly damage the organization of divisions in combat and gives a combat bonus to the divisions on their own side. High amounts of CAS on your side means the enemy can't attack or defend as long as normal. In small amounts, it's nothing to worry about, but an enemy who uses plenty of CAS can make enemy Air Superiority even more of a problem. Luckily, the AI isn't too smart. But against players, CAS spam is common.

Combat Width
Size of the division and size of the battle. Default battle width is 80, +40 per extra attacking province. This is why it's important to have divisions of sizes 10, 20 and 40.

Command Power
Used to promote generals, to make them learn traits and to activate operations for your army. Not that important.
How to improve: High war support, certain doctrines and national focuses.

Compliance
Dictates how much resistance there is in a state and how many factories you can use there.
How to increase? Ensure that your garrison is sufficient and...wait. Spies can be used to setup collaborative governments in foreign countries which gives you a large compliance gain when you occupy that state. Spies can also be used to lower resistance.

Defense
Each defense gives you a fairly high chance to evade one attack. Much like with piercing and armor, it's not necessary to have more defense than an enemies soft attack (or hard attack in a high hardness division). If you have 150 soft attack, and they have 100 defense, ignoring hardness, 50 of your attacks are guaranteed to hit. So if your defense is way higher than the enemies attack, it might be smart to shift some of that defense over to soft attack to do more damage to attacking divisions instead.
How to increase? Pretty much everything gives a tiny bit of defense, but the best are, like with org, inf/mech battalions. Engineer companies are also quite strong on defense.

Entrenchment
The ability for a division to 'build defenses' before a battle. Builds up when a division is inactive. Adds a significant defensive bonus. If the division moves, all entrenchment is lost.
How to increase Engineer Support

Eq. Capture Ratio
After every battle a certain amount of equipment is lossed/captured. This is separate to the equipment you gain after a nation capitulates. This stat can be improved with maintenance support and isn't very important.

Exp. Loss
Pretty self explanatory. Lowers the rate at which divisions lose experience in combat. Experience is lost when manpower is lost and reinforcements were made. However since experience is also gained through combat, this really means your divisions will gain experience faster. Meaning faster veterans and significant combat boosts.
How to improve? Field Hospitals. As a result, Exp. loss goes hand in hand with 'Trickleback'

Fuel
Very important resource. Especially if you have a large navy/airforce, though if you rely on tanks it might also become an issue. You gain it from oil which you can trade for, extract or make with a synthetic refinery. It's also important to know that on the far right of the industry research tab you can research better oil refining methods which effects ALL oil sources. Gaining you more fuel per oil. Closing your economy can also alleviate some stress. It's very easy to forget about, but the effects of running out of oil are catastrophic. Divisions fight at 10% strength, planes are 75% less effective and ships fight at half strength.

Factory Output
A percentage bonus to the output of factory production.
How to increase? Industry research, trade law, stability.

Fuel Capacity
Not that important. Dictates how long a division can operate at full fuel capacity without a fuel supply. Only necessary to pay attention to for amphib's.
How to improve? You can't.

Fuel Usage
How much fuel your division consumes. This plays into your fuel capacity but it's more important in regards to your nations fuel supply. If you are low on fuel, you might need to improve your divisions fuel usage.
How to improve? Logistics companies.

Garrison
The divisions/manpower diverted to controlling your nations resistance growth. There is also a 'Damage to enemy garrisons' modifier that can be sometimes gained which results in more losses to the enemy garrison when they occupy your states. Not exactly a game winning modifier however...
How to improve? see 'Suppression'

Hardness
IMPORTANT
To put it simply, hardness is the percentage of a division/battalion that rejects soft attack for hard attack. So for example, with 0% hardness (pure inf division) hard attack never matters. For 100% hardness, soft attack never matters. Soft/Hard attack is only relevant in regards to hardness. Hardness is NOT shown typically with the other stats, it's shown via a bar inbetween the adjusters and the base stats.This 1936 German Light Tank has around 56% hardness. Adding a heavy tank division to a pure inf division can add around 10% hardness to a division, already significantly decreasing the amount of soft attack damage taken. Extra note: when resistance grows high enough your garrison takes manpower and equipment losses, but this is lessened by the hardness of the garrison. Because org isn't necessary, you can make a high hardness garrison to lessen the losses suffered due to high resistance.

Hard Attack
See hardness. Hard attack hits the hard part of the division when soft attack isn't applied.
How to improve Anti-tank, tank destroyers.

HP
Is your divisions 'second health bar', and is indicated by the orange bar on each of your divisions. When your HP reaches zero, your division is killed. HP recovers over time, and is made up of both manpower and equipment. If supply is bad or if the division is encircled, the division will not recover any HP and will be easily killed. This is the main reason why encirclements play such a large part in HOI4 combat, as it's the easiest and most common way to entirely remove a division from play. Organization drops faster and is more relevant in combat but unlike organization, HP effects how strong your unit fights in combat. 50% HP results in 50% fighting strength. So if you orange bar is low, you should fix it, fast.
How to improve? Same with org, inf, cav, motorized, mechanized. And much like with org, tanks/arty are very poor when it comes to HP. All things considered though, HP is only something that should be cared about when it's not recovering to full or when your division is planned to be out of supply, i.e paratroopers and naval invasions. Org is more relevant to how combat plays out.

Ideology
The type of government your nation has. To explain the differences... (In the GAME)
Democracy: Sucks. Don't play it unless your going after achievements, roleplaying, or are playing competitive multiplayer. They can't even justify war goals. A bit better with a good focus tree. The only real 'benefit' they have is they can guarantee everybody, regardless of ideology.
Communism: Can justify but it lacks some of the benefits Fascism has. If you prefer the focus path, go for it.
Fascism: -80% justification time when at war with a major power... Declare war on a random one somewhere (preferably without a navy) and have fun taking over everybody around you quickly. Practically the only way for world conquest before the game grinds to a halt due to lag in the late game.
Non-Aligned: Worse than both Communism and Fascism as they require 50% world tension before being able to justify. However, they usually have fairly fun focus trees that allow them to go to war against a few nearby nations and 50% tension limit is still better than being a democracy.

How to change: Usually through the focus tree and ministers. Decisions are also important: You MUST use a decision for referendum to change ideology peacefully at over 50% popularity. Otherwise, civil war might occur. Ideology can also be externally boosted with spies.

Initiative
Speeds up battle planning. More importantly increases the chance a division joins combat. Pretty much a must have as divisions joining combat will ALWAYS be useful. Separate from tactical initiative. See 'tactics'
How to improve: Signal companies

Maximum Speed
The max speed that battalion can go. The divisions max speed is dictated by the slowest battalion, so even if you have one infantry battalion in an entire motorized division, they will move as slow as that infantry. This is why motorized/cav/mechanized is necessary to go with your tank battalions. Though heavy tanks aren't slowed down much by inf.`
How to improve? Make a variant with an improved engine. Keep in mind though that motorized/mechanized can't have variants and is required to add org to your tanks and armored cars. So even if you have a light tank capable of going 16-17km/h, they will be brought back down to ~12km/h, so there is no need to improve the engine of light tanks.

Nukes
Nukes in HOI4 aren't game changing but they are still useful. Dropping one on a state requires air superiority and a strat bomber in range. The effects of the nukes are: Significant damage to everything within the state. So infrastructure, factories, air bases, AA, forts, that sort of stuff. More importantly, significant damage to the 'HP' of units, making them fight at a weakened strength. Nukes are primarily used late game along an entire front, when naval invading, or when crossing rivers, breaking the enemies defensive lines. The damage done to HP is a result of destroying both the manpower and equipment of the division. Nuking enemy tanks frequently will put strain on their tank stockpiles.

Organization
The 'health' of your divisions. Indicated by a green bar for each division. It is one of the primary factors dictating how long your division can fight for before retreating in both defense and offense. A division with severely damaged org will move drastically slower than fresh divisions. It does NOT alter the effectiveness of that division in combat however, other than how long they can fight for.
How is it lost/lowered? Org is lost when training, when moving via strategic redeployment, a small amount is lost from moving into enemy territory, from enemy air and from ground combat.
How to improve a divisions base org: Certain battalions in the division designer will add org, others will lose it. Inf and mobile inf tends to increase it, tanks, artillery and other specialized battalions tend to lower it.
Org can not be lowered below 10 but that is not enough to make a competent division with. This is why pure tank and pure artillery divisions don't work well. It's a balance between high damage values and high org.
How do I regain org once I lost it? See 'Recovery Rate'.

Piercing
See 'Armor'. If higher than the enemies armor, the enemies armor becomes irrelevant.
How to improve: Anti-tank, Tank destroyers.

Political Power
Used to change government laws, employ advisors, recruit generals, activate decisions, guarantee independence, etc. Default gain is 2 a day, doing a national focus costs 1 per day.
How to improve: Your leader can give a bonus, certain advisors like the silent workhorse, certain national focuses, by not researching a national focus. By having high stability.

Production Cost
Self explanatory, the cost something costs to build. Important to pay attention to this if your nations industry isn't fantastic. For example, heavy tanks cost around three times as much as light tanks to produce.
How to improve? Don't be greedy.

Production Efficiency
How much of the military factories production you make use of. They if they provide 4.5 production a day, at 50% efficiency, your military factories give 2.25 production a day.

Production Efficiency Base
The amount of production efficiency a new production line starts with.

Production Efficiency Cap
The max amount of production efficiency you can have. Capping at 100%, obviously.
How to improve? Researching tools.

Production Efficiency Growth
How quickly your production efficiency grows to reach the cap.
How to improve? Researching streamlined line.

Production Efficiency Retention
How much production efficiency is saved when your military factories change production of an item.
How to improve? Researching dispersed industry and flexible line.

Puppet (This relates mostly to the 'Together for Victory' DLC)
A nation that is under your control, but has its own government.
Benefits: Allows you to train divisions using another manpower pool. This is practically a necessity if playing as a minor nation. Puppets will also give you their own divisions, but they usually suck. It's better to train your own using their manpower.

More factories from poor, very small areas. For Example, Africa. As the puppet will get the default focus tree, they can do focuses that give them factories and building slots. Allowing even areas with no factories the ability to develop.

License production puppets equipment.

RESOURCES If you are in dire need of a resource; puppet, do not annex. Firstly, due to resistance/compliance, how much resources you can access is drastically decreased in comparison to a puppet of yours. Secondly, it allows you to change your trade laws. You can benefit from free trade while importing all of your resources on the cheap. Trading with your puppet is significantly cheaper (1 factory for 40-80 resources, not just 8).

The puppet keeps the navy of the previous nation. By building in their territory, you can reduce their autonomy to then annex them. Giving you full access to their navy.

NO GARRISON. Japan pretty much has to puppet a lot of China as the manpower requirements for maintaining the garrison are quite costly. Even areas with 100% compliance need some sort of a garrison so sometimes it can be better to puppet rather than annex, particularly if you're expecting a war soon.

Negatives: Less factories. While puppets can send you some to use, you gain far more from annexing.

They spam you with volunteers you don't want. You think your much needed tank division is done, while really your rubber puppet over in Asia just gave you a 2 width cav division. You can't switch it off.

Like with all AI 'allies', they can flood your frontlines. Leading to supply issues. You can control their divisions to force them to garrison their homeland however, but you'd need to keep doing it.

They gain autonomy by trading with you, giving you manpower and factories, etc. Which means you either have to keep fighting their autonomy growth, or they will become less effective over time as they reduce their puppet level. They can even free themselves from you.

Most importantly. Less cool. But you can always annex them later.

Recon
The higher your recon, the more likely your commander will get the tactical initiative. See 'tactics'.
How to improve: Recon support companies.

Recovery Rate
All divisions regenerate some org while not in combat at a fixed rate, the recovery rate is an addition to that regeneration. This recovery rate is lost damaged when out of supply.
How to improve: Certain support companies like recon, certain battalions like motorized. But much like org, is lowered by other support companies and other battalions.

Reinforce Rate
Determines the speed at which your divisions join a battle from reserves.
How to improve: Researching radio, and certain doctrines. By raising 'initiative' through signal companies. Certain military ministers.

Reliability
Higher reliability results in less equipment lost during combat. Particularly helpful for expensive equipment such as heavy tanks.
How to improve: Maintenance Support companies or by making a variant with upgraded reliability.

Soft Attack
The damage done to the 'soft' part of an enemy division (See 'Hardness'). As a simple rule, good vs infantry, pretty good vs low level motorized, bad vs tanks and high level motorized.
How to improve: Artillery, particularly rocket arty.

Stability
High stability results in production and political power bonuses. Low stability does the opposite. The main problem with low stability comes in the form of mid-war crisis. All of the crisis are terrible for your country, and the only way to stop them once they happen is to spend a massive amount of political power which is what's needed to improve your country's stability in the first place. Remember you lose 30% stability when at war.
How to improve: Certain advisors increase stability, as do certain focuses and events. The only way to repeatedly improve stability however is through decisions.

Supply use
Amount of supply consumed per day. Uses significantly more than base when doing stuff like moving and being in combat, uses slightly less than base when inactive.
How to reduce: Logistics companies, smaller sized divisions. 40 width tanks might be great in Europe but are probably not a good idea in South American jungles.

Suppression
How efficient a division is at suppressing local resistance. To explain, due to the recent resistance update, a garrison is automatically applied to every state suffering from resistance, with a template design of your choice, consuming both manpower and equipment in the process. The higher the resistance, the more suppression is required. So having a high suppression template in your garrison will drastically reduce both manpower and equipment expenditure in the garrison.
How to improve? Use military police, cav, motorized, light tanks and armored cars. Armored cars with military police are probably the best (see hardness) but VERY costly, don't use them unless you have plenty of either armored cars or factories spare. Best budget garrison is probably cav with military police. Also, the org of your garrison doesn't matter, so a pure light tank/armored car division is fine.

Tactics
Each tactic gives a variety of combat modifiers. They are chosen every 12hrs. The side with the initiative can counter the enemies tactic, preventing it from having an effect. In pretty much all circumstances, the side with the higher recon value has the initiative. So...
How to improve? Recon support. If you have the same level of recon -> The level of your leader.

Trickleback
VERY IMPORTANT (If you struggle with manpower)
This is the percentage of troops that are returned to your manpower pool that would have otherwise died. For example: A normal division might lose 100 men in a battle. If that division had 20% trickleback, 20 of those men would survive. This significantly lowers the amount of people that die in combat. Max trickleback is 50%, meaning half of your men who would have typically died, would survive. This increases how long you can go before raising your mobilization law and is a necessity for smaller nations.
How to improve? Field hospitals. Connected to improving 'Exp-loss'.

Weight
Not too important. Determines how many convoys/transport planes are necessary to transport a unit. Usually it isn't something to worry about, but if you are low on convoys it might be down to sending 24 40 width heavy tanks over to Europe.