There are 3 types of physical damage, 4 types of primary elemental damage and 5 types of combined elemental damage. And there are 3 basic factions: Grineer, Corpus and Infested. Each faction has its own resistance and vulnerabilities, based on their health classes.

(In this section I won't cover the Corrupted (Orokin), which is a mix of every faction (so there's no simple way around), and Wildlife, which usually is not a challenge to any weapon.)

Before applying modding principles you should know which damage type is good for each faction and which is not.

A more comprehensive explanation is available on Warframe Wikia[warframe.wikia.com]. Some damage types are good for more than one faction.

1. Grineers - Puncture (Weakened) - Viral (Virus) OR Corrosive (Corrosion) OR Radiation (Confusion)

• Weakened = 30% less damage over a duration of 6 seconds
  
• Virus = reduces maximum health to 50% for 6 seconds. Viral = Cold + Toxin
  
• Corrosion = permanently degrades the target's current armor by 25%. Corrosive = Toxin + Electricity
  
• Confusion = affected enemies target and can be targeted by their allies for 12 seconds. Radiation = Heat + Electricity

NB! Radiation damage is good (+75%) against the alloy armor = bombards, elite lancers, eviscerators, napalms. Corrosive damage is good (+75%) against the ferrite armor = every other Grineer + Juggernaut. However my recommendation is viral damage (if possible), because it is good against ALL cloned flesh = Grineer. Besides, viral damage is also good (+50%) against the desert skates, ferral kubrows and drahks, although corrosive is better (+75%).

2. Corpus - Impact (Stagger) - Magnetic (Disrupt)

• Stagger = knocks down the enemies leaving them incapable of fighting for a while
  
• Disrupt = reduces the target's current shields and maximum shield capacity by 75% over a duration of 4 seconds. Magnetic = Electricity + Cold

NB! Viral damage is also useful (+50%) against the Corpus flesh, but magnetic damage is better (+75%) and is good against basically all shielded Corpus enemies.

3. Infested - Slash (Bleed) - Gas (Poison Cloud) OR Blast (Knockdown)

• Bleed = 35% of your weapon or power's base damage per 6 seconds as finishing damage. Therefore, this damage bypasses shields and is not affected by armor or enemy resistances
  
• Poison Cloud = poison effect to all enemies within 5 m of the struck enemy. Poison deals 50% of your weapon's base damage per 8 seconds
  
• Knockdown = stunns all enemies within 5 m of the target

NB! Gas damage deals +50-75% against small Infested + Juggernaut. Corrosive damage deals +75% damage against big Infested (mutalists, ancients, brood mother and boiler). Therefore the best option probably is to have a melee with gas damage to slash through the Infested masses and a primary corrosive weapon against less common ancients and those nasty mutalists and brood mothers.

Finisher damage is a special type of damage that is particularly effective against Phorid.

1. Always increase the base damage

Always use the following mods:

• Rifles/Bows/Sniper Rifles: Serration, Split Chamber
  
• Shotguns: Point Blank, Hell's Chamber
  
• Secondary: Hornet Strike, Barrel Diffusion
  
• Melee: Pressure Point

If you have prime versions of those mods, use them!
And always use maxed out stance mods for melee weapons to increase the mod capacity, preferably, of the same polarity.

2. Build upon the weapon's strenghts

• What is the strongest type of its damage?
  
• Does it have an inbuilt elemental damage?

Examples (not necessarily the only way)

• Dread's strength is high critical damage and chance =>Vital Sense.
  
• Quanta's strength is an inbuilt electricity damage, so, apply Stormbringer and combine it with the Cryo Rounds – you'll get a high magnetic damage, that is useful against the Corpus faction.

UPDATE! As other players have pointed out in the comment section, elemental damage mods are always better than physical damage mods (e.g., Sawtooth Clip, Piercing Hit), because, in total, elemental damage mods provide more DPS than physical damage mods. I previously wasn't careful enough and thought that this is not true for weapons with high base physical damage (e.g., Mitter, Daikyu, Dread), but after another check, I must agree that elemental damage mods are better and the difference is substantial (500, 800 damage points or even more). This means that physical damage mods are completely worthless and, probably, the only reason to use them is, if you haven't got those extra two capacity points (9 vs 11 for max rank).

This also changes how you should apply the second principle. You won't be building on physical damage strength's anymore. However, it still can serve as a guide for specialisation (see the 5th principle).

• For example, if a weapon has a high puncture damage, add Radiation or Viral damage and use it as an anti-Grineer weapon.

NB! Enhancing the critical damage, imo, is only worthwile, if the critical chance is at least 20%. Enhancing channeling efficiency is only worthwile, if you actually want to spend the energy on channeling instead of spending it for powers.

3. Compensate the weaknesses

Slow reload speed? => Fast hands
Slow fire rate? => Speed Trigger (Speed Trigger usually is a good idea, because it means more damage per second)
Short melee? => Reach (swords, daggers, machettes,...)
No need to use Reach for long swords, nikanas, poles and hammers as they are long enough.
Also, brawler weapons (e.g., gauntlets) are so short, that even Reach would not make a difference. So, in case of gauntlets use something else.
Slow hits? => Fury (use it even if you have sufficient speed, if you have the capacity)
Low magazine and/or total ammo capacity? > Imo, the ammo/mag capacity mods are waste of space, because there's plenty ammo laying around and for those weapons that run out of ammo quickly (e.g., Phage) you should use ammo convertion mods instead (e.g., Shotgun Ammo Mutation, Sniper Ammo Mutation). Besides, for example Magazine Warp, usually does not provide a substantial improvement anyway.

NB! Definitely don't use physical damage mods for weapons that do not have a physical damage (e.g., Quanta). Don't use critical damage mods for weapons that do not have a critical damage (e.g., Seer, Miter).

4. Prioritize

You can't fit everything. Decide what is most useful for you

5. Specialize

• Use configurations for various modding options, if it's your strongest weapon or you are currently ranking it up on various planets. For example, Quanta was my strongest weapon for a long time. Therefore, I had modded it for radiation damage against Grineers, magnetic damage against Corpus and corrosive damage against Infested. The latter combined with Dragon Nikana as gas slasher.
  
• Have different good weapons against different factions. For example, I primarily use Quanta as my anti-Corpus weapon and Phage as my primary anti-Grineer weapon.
  
• Also, when specializing, use the faction mods (e.g., Smite Grineer, Bane of Corpus). +30% damage ON TOP OF EVERYTHING is a lot!

• Anti-Grineer combos: viral + cold (cold provides +25% damage against bombards, napalms, elite lancers and eviscerators) OR EVEN BETTER – Viral + Radiation
  
• Anti-Corpus combos: magnetic + toxin (toxin provides +50% damage against non-proxy Corpus (humans), although robotics have -25% resistance)
  
• Anti-Infected combos: gas + electricity (cold provides + 25% damage against Phorid, but other infested are resistant against cold, therefore it's better to use electricity that has no bonus, but also no resistance)

1. The weapon has a high total damage (at least 1000 at rank 30 and when modded with the basic mods) and there are no better alternatives of the same type
2. It has no serious drawbacks (for example, Daikyu has a high Puncture damage, but unfortunately it is quite impractical due to its mechanics)
3. There are slots that need polarization
4. You like to use the weapon often

The Corrupted (Orokin) have different health classes. So, weapon configuration might be tricky.

Here is a one way.

1. Primary weapon with a viral damage – good (+75%) against the corrupted lancers, which are very common in the Void, as well as the more challenging heavy gunners and bombarders. It also provides an advantage (+50%) against the corrupted crewmen, which are also very common in the Void, and nullifiers. Theoretically, viral damage combined with a high puncture damage would be even better (but I don't know if there's such a weapon). If you have the capacity, add electricity as a 3rd element against MOAs and drones.

2. That's the easy part. You still need magnetic damage against fusion MOAs and drones (also good against corrupted crewmen and nullifiers) AND corrosive (+75%) or blast (+50%) against corrupted ancients. You can cover those two elemental damages with secondary and melee weapons – how exactly – that's up to you. Take into account that Mag can help against robotics (especially the drones) with his powers, but I'm not saying it's the only way.

Besides, there are other considerations, in terms of warframe choice (protection against damage, crowd control, team support etc.). Auras matter a lot, too. But that's another story.

Warframe Builder[warframe-builder.com]: modding examples for weapons, warframes and companions, weapon comparison/building

Warframe Utility[play.google.com]: Android application that includes alerts and damage calculator (among other things)