First, we need to know what exactly a Combat Engineer is. A combat engineer is an engineer who generally plays much more offensively than a regular Engineer, heading directly into combat. Most of the time, Combat Engineers will use the melee weapon known as the Gunslinger. The Gunslinger provides Engineers with an additional 25 maximum health and replaces their standard sentry with a mini-senty.

Mini sentries are very weak, weaker than even a level 1 sentry, and they can't even be repaired, however, mini-sentries only require a mere 3 seconds to be fully deployed and begin firing on your team and they only cost 100 metal, allowing the engineer to deploy 2 different ones without even going to get more metal (he can still only have 1 deployed at a time though), allowing him to spam mini-sentries as he pleases. While a sentry that is weaker than even a level 1 sentry may not sound like much of a threat, the main threat of it comes from the fact that many people assume it's a lot weaker than it actually is. It can do 48 damage per second, killing a fully healed Heavy in 7 seconds, and a fully healed Soldier in 5 seconds. The fact the sentry can't be repaired, means the Engineer can simply just put down a mini-sentry anywhere and not care about it until it's destroyed.

Soldiers and Demomen are still the most effective at taking out these sentries. Their high HP, splash damage, and high damage output lets them destroy the sentries in a matter of seconds. A single Soldier or Demoman sometimes isn't enough to take out a Combat Engineer though....

The Vaccinator is a bit of a strange Medi-Gun. Many people give up on using it because it seems pretty complicated to use, but it's actually relatively simple.

The most simple stats are the 66% decrease in overheal build time and 50% increase in ubercharge rate. Pretty self explanatory.

The Vaccinator can also provide a 10% passive resistance to the following damage types: Bullets, Explosives, and Fire. Rather than just providing a 10% passive resistance to all of them at once, you have to cycle through them manually by pressing your reload button ("r" by default). Not only will your healing target gain a 10% damage resistance to the selected type, but you as well. You also get healed for 25% of the damage dealt to your healing target, assuming you have the correct resistance matching the damage type that was dealt. What the game DOESN'T tell you about this passive resistance is that it grants you a 100% critical hit resistance against the selected damage type, making it incredibly broken against snipers, but that's a story for another time.

The ubercharge of the Vaccinator is also a bit strange. Rather than having 1 big 8 second ubercharge, you have 4 small 2 second ubercharges, which can be done seperately by simply pressing the secondary fire button once for every charge, or can be done all in one go by holding it down until it's over. Every mini ubercharge represents 25% charge. The fast charge time of this Medi-Gun lets you pull off a 2 second charge about every 8 to 10 seconds.

The best feature of this Medi-Gun is its' ubercharge. When you activate its' ubercharge, you gain a 75% resistance to the selected damage type. That may sound underwhelming, but considering how fast you can pull off ubers, and how you can effectively stop them whenever you want, it's a pretty fair trade-off. Don't underestimate how much damage reduction 75% may sound like....
(Don't forget you're also healing your target at 24 HP per second if he's constantly taking damage)

Rocket damage at point blank range: 112 becomes 28
Fully charged Sniper Rifle shot: 150 becomes 37.5
Engineer's Shotgun at point blank range: 90 becomes 22.5
Mini-Sentry DPS: 48 becomes 12

Wait a second....At the bare minimum, you heal people for 24 HP per second, and Mini-Sentries can only do 12 damage per second when you're ubercharged? That means your ubercharge can completely negate mini-sentry damage to your healing target and still have a 12 HP per second healed left over!

You may have already figured it out, but in case you haven't, to counter Combat Engineers you have to use the Vaccinator and pocket a Soldier or Demoman. Just make sure you know how to deal with each of the Engineer's other weapons....

Primaries:

Shotgun: Nothing special you have to worry about. Just pop ubers if you feel in danger.

Frontier Justice: Since the Vaccinator will provide your healing target and you with a 100% critical hit resistance, it's a direct downgrade from the regular shotgun when he's dealing with you.

Widowmaker: This'll let him put up more mini-sentries, and effectively gives him infinite ammo if he lands all of his shots. Deny the Engineer his infinite ammo and metal by popping your uber constantly when dealing with him. It will cost him 30 metal to fire it every time, and with your uber active the most metal he can get back is 23 per shot.

Pomson 6000: More intimidating than it actually is. If he hits you, you can easily restore that 10% ubercharge lost in about 1 or 2 seconds, so it's not a major loss ubercharge wise. He also can't decrease your ubercharge if it's currently active. Pop your uber as soon as possible and hope your buddy can deal with him as soon as possible

Rescue Ranger: This let's the engineer pull off some fancy tricks by transporting his mini-sentries everywhere. Too bad you can essentially completely negate them. The low damage output also makes it harder for him to kill you and your healing target.

Secondaries:

Pistol: Pretty basic, not much to worry about.

Wrangler: Gives the Engineer's mini-sentries a DPS of 24 which you can still negate, and triples their health. Just pop your uber and the engineer is pretty much helpless unless he changes to a different weapon.

Short Circuit: The main reason I would reccommend a Soldier over a Demoman for his secondaries that aren't projectile based. If you really have to, then have the Demoman or Soldier get up in his face with their melee weapons.

Melees:

The Gunslinger: It can't have random critical hits, so you don't have to worry about instant death, but if the Engineer lands 3 hits in a row, the third will always be a critical hit. Since you don't have a melee damage resistance, watch out for Engineers that will just charge at you or your healing target with the Gunslinger.