Her name says it all. Yes, her. Valve has repeatedly dropped subtle and not-so-subtle hints that the Pyro is in fact a woman, so from hence forth she shall be referred to in the feminine. Our mumbly firesmith is superb in close combat, and can throw an entire team into disarray with a well-timed ambush. The Pyro is also the best anti-Spy class in the game. In the pre-update days of Team Fortress 2 the Pyro was a bit underpowered, much like the Scout. However, her updates have made her much more well-rounded, especially from a defensive perspective.
The Pyro's flamethrower excels in sudden close-quarters attacks. A lone Pyro can do an incredible amount of damage if she can get the drop on the enemy, and can even take down a sentry or a Heavy if she's quick enough. Even if the Pyro can't finish the kill, getting set on fire is often a death sentence for the victim, especially for low-health classes like the Scout or the Sniper. It's for this reason that I actually like the Backburner as opposed to the normal flamethrower in some situations. The auto-crits from behind can be devastating if the Pyro can get behind a large group pushing the cart or capturing a point. It's also effective chasing down someone running away, since their backs will be deliciously faced towards you.
On offense, the Pyro can also be extraordinarily effective at pushing forward, making her an excellent choice for the normal Medic uber. Pyros run at "normal" speed, giving them an edge over Demos and Heavys in this regard. Demos are still better for taking out sentries, but the speed advantage means Pyros can attack a wider range of targets in those precious ten seconds of uber. Plus they can adjust strategy mid-uber if they need to and still be fast enough to make it count.
Speaking of ubers, the addition of the air blast in the Pyro update has made her the best anti-uber weapon in the game. Pushing a ubering Medic and his ward back even a little bit wastes a few of those ten seconds, and often the Pyro can juggle the Medic in the air for the entire duration of the uber. Air blasts can also clear the Demos sticky bombs and actually reflect Demo grenades and Soldier rockets. Doing damage with this tactic is tricky, but the reaction of a Soldier killed by his own reflected crit rocket is always priceless.
A single Pyro can throw a whole team into chaos. Setting five or six people on fire not only does damage, but it also greatly disorients them. The burn effect "bumps" the victim's aim every second, decreasing their accuracy. The group will also be forced to slow down and heal any damage that they've taken, and if the Pyro can take out their Medics she'll probably get at least a few kills. Medics can put out everyone else but not themselves, and the rest of the team will likely rush to their aid if they get lit up, throwing them even further out of whack.
Again, this is where the Backburner can come in handy. Sneaking in behind a group with it will usually result in at least one kill, hopefully their Medic. The ensuing chaos will also likely result in some more backs to burn.
Spy check. Fucking. Spy. Check.
Pyros are the most effective anti-spy deterrent in the game, and thus it pays off to be paranoid. Once set on fire the Spy's disguise and cloak become useless, forcing them to run away. It's is 100% worth it for a Pyro to spy check anyone that looks even the slightest bit suspicious. Even randomly flaming the nooks and crannies of the map where Spies might hide get her a catch every once and a while. A Pyro that is willing to go on Spy patrol duty while her team's Engineers are setting up is line for some sexual favors after the game, maybe even during the game.
Don't tell me that flare was standard prison issue…
I like the Flare Gun. It isn't a very popular weapon from what I've seen, but it's the Pyro's only true long range attack. As I said before, the burn effect is very disorienting for the victim, so hitting someone with a flare from far away will likely make them retreat unless they have a Medic with them. This can be especially devastating to Snipers, as they will have to back off and they don't have the health to burn for very long. The flares fire in an arc so aiming it can be tricky, but it's well worth it to get it down. Outside that, the shotgun is effective enough at finishing off burning enemies, especially faster ones that can get away from the flamethrower.
The Axtinguisher seems to be the usual melee weapon of choice for the Pyro, and with good reason. The guaranteed crit when the target is on fire enhances her close-up skill even further, especially against slow, high-health enemies like the Heavy and the Soldier. The Degreaser's fast weapon switching makes this strategy even more effective. However, I usually stick with the Homewrecker. Being able to knock Spy sappers off of Engineer buildings is invaluable, as often I will kill a Spy right after he backstabs an unfortunate Engineer and puts sappers on everything. Saving those buildings can be the difference between victory and defeat, and I'll take that over a chance to take down a lone Heavy any day.
As one might guess, with the Pyro's close-range skill comes long-range deficiency. With the small exception of the Flare Gun, none of her weapons are suitable for long-range combat, thus Pyros need to avoid wide open fields when they can and stick to small corridors and trenches where their effectiveness is maximized. She's an easy target for Snipers, and Soldiers and Demos can also deal with Pyros fairly easily if they can see her approaching. A few grenades/rockets are enough to stop her cold, and even the air blast can't defend against a sustained explosive onslaught.
The Pyro's close-range prowess is useful in almost any map. She can attack with the best of them, and the addition of the air blast makes her invaluable on defense as well. Spies cower at the sight of Pyros in any situation, making an alert Pyro crucial to any Engineer base setup. She's similar to the Soldier in her offensive and defensive usefulness, albiet with a vastly different skillset. Just keep her out of the open range and she'll be your flame retardant heroine more often than not.