How to not look like an idiot in Team Fortress 2: the Pyro

When playing as him:

-Ambush

The Pyro is best for hiding in corners and behind doorways, waiting for feeble-minded enemies to run by so he can tickle them to death with napalm. The flamethrower is (unsurprisingly) a short-range weapon, so hiding and ambushing like a guerilla is really the only way to get good results as the Pyro. 

-If they're too far away, they're too far away

There's nothing quite as adorable-yet-tragic as the sight of a Pyro, flamethrower blasting, running after a faster character class who is not only backpedaling and blowing the hell out of the Pyro with secondary weapons, but is also juuuuuust out of range of the fire. Trust me, guys; the guy you're chasing isn't suddenly going to decide to stop moving backwards (unless he isn't facing you) so you can finally tag him with your flamethrower and set him aflame.

If he's too fast, then he's too fast -- it's not a big deal, just run away and find someone else to torch. That, or switch to your shotgun, which is a good weapon in its own right.

-Use your anti-spy device

Otherwise known as a flamethrower. Since roughly 99.9% of all TF2 servers don't have friendly fire enabled, the Pyro cannot possibly set any of his own teammates on fire. He can, however, set disguised spies on fire. Recklessly firing your flamethrower at every team member you see may initially seem like a dickhead move, but it's actually an invaluable strategy for unearthing and killing Spies.

-Setting people aflame isn't an immediate thing

Just because you quickly graze your flame arc over a baddie for a few seconds doesn't necessarily mean you've actually managed to set him on fire. Listen for the WHOOOSH and make sure your target has flames on his body even when you aren't aiming at him before you decide to move on -- but then again, it's probably a better idea not to move on at all if the guy is still kicking.

-Hang out at capture points

As the Pyro is a powerful close-range fighter, he's immensely useful at defending capture points. All he has to do is stand near a capture point and torch any- and everyone who attempts to capture it. Unless you die and a new enemy comes along whom you don't have the opportunity to set aflame, this makes it literally impossible for anyone to capture a point; they'll either die immediately due to the immense flamethrower damage, or they'll continue to lose health as the flames eat away at them, forcing them to either leave the capture point to find health, or to stay still and die, leaving the point uncaptured anyway.

-Your flamethrower doesn't work underwater

This isn't really as big of a problem as one might think (it only applies to cp_well in any significant sense), but remember that firing your flamethrower whilst submerged will only result in harmless bubbles coming from the barrel -- though you still lose ammo.

-The power of fear

One of the Pyro's main strengths is his ability to essentially close off entire areas by standing and spouting flame at his enemies. If said enemy is of a class which needs to get in relatively close to do a lot of damage (like a Scout), then the enemy is probably going to have to run away and choose another course. Hell, even if the Pyro encounters a combat-adequate class like the Soldier, the spouting fire of the flamethrower will still probably force the Soldier to retreat whilst firing. 

This makes sending a Pyro at a group of enemies, while useless if you actually want to kill them all, a very intelligent strategy for slowing down and distracting opponents. People are terrified of being set on fire, because it represents extra work and a waste of time; you'll either die and have to wait, or you'll live and have to either throw yourself into a body of water, or retreat and refill your health. A Pyro can run into a crowd of baddies and flame them, whilst his supporting team members blast away at the panicked and confused mob.

When playing against him:

Also, keep your distance. It may often be impossible to do so (especially if you get burned to death before you can actually run anywhere), but try to stay the hell away from the Pyro if at all possible whilst hitting him with ranged attacks (salami_tsunami suggests rockets and grenades; Jeff says that a fully-charged Sniper shot results in a Pyro instakill).

Eschatos gave me some great advice if you're a Demoman; since Pyros tend to hide behind doorways and in corners, always try to bounce grenades in likely Pyrolicious areas so as to smoke him out and potentially kill him. As Eschatos says: "The Pyro's only real powerful strategy is ambushing, so naturally you want to deprive him of all oppurtunities."

That's really all I've got, unfortunately -- from bad astronaut to Eschatos to my friend Jeff, the only real non-class-specific anti-Pyro strategies we could come up with involved either putting yourself in a position to be healed, staying the hell away and shooting at him, or diving into water whenever possible. The Pyro is a rather simplistic character, and as such requires rather simplistic methods of dealing with him. His power level has frequently changed throughout the course of the beta, as well; he used to be weak to the point of uselessness, then he could kill people just by looking at them, and now he's somewhere in between. Learn the amount of damage he can do, and prepare yourself for it.

If you've got any anti-Pyro strategies you'd like to share, or if you think he isn't actually all that overpowered, then I urge you to hit the comments.