team-fortress-2tf2-pyro
On the Detonator's wiki page, it says the following under base damage:
I know that the 30 damage is achieved when the detonator is a direct hit, and the 10-20 damage is achieved when the flare is manually detonated (the damage depending on the distance from the detonated flare to the enemy), but what is a flare explosion?
The wiki page provides no more information than simply "flare explosion: 56 [damage]", so what is a flare explosion, and how do I use one on an enemy?
The compression blast range appears to be slightly shorter than the range of your actual flamethrower, from personal experience. You can compression them anywhere within that range and you'll reflect. The angle that it gets reflected back at is dependent on where your cross-hairs are aimed when you hit the compression blast.
Additionally, for grenades, there appears to be some determining factor with return velocity depending on your distance from them. I can't find a lot of info about that though.
Rockets will travel indefinitely until they hit an object. Arrows seem to function similarly to grenades, but less noticeably so due to their natural high speed and lack of "bounce".
In regards to lag compensation, the compression blast is NOT lag compensated. You're going to have a harder time reflecting rockets / grenades / arrows if your lag is higher, especially when dealing with the higher velocity ones like the Direct Hit rockets and arrows.
Edit: As per the link in the comments, edited the above to indicate that the compression blast is NOT lag compensated.
It's hard to say "when" you should compression blast. It's a function of distance and velocity, rather than time from fire. With any projectile, it's nearly impossible to reflect at short range without some sort of luck / godly reaction speed. At longer distances, the Direct Hit and arrows can still be relatively difficult to reflect. Again, it's about the distance, not the time before it hits. The best way to get a feel for it is to play it and learn it intrinsically, not go by a paradigm that there's a hard rule for timing it based on seconds.
You can reflect:
Soldier Rockets (all types)
Pyro Flares (all types)
Demoman Grenades (all types)
Demoman Sticky Bombs (Sticky Bomb Launcher/Scottish Resistance)†
Huntsman arrows (which can headshot enemies) 
Sandman Baseballs Wrap Assassin OrnamentsWhen you reflect projectiles they are promoted to minicrits (if they weren't), change sides with your team and ownership to you (with the exception (†) of stickies). This means, however, that they will still do splash damage to you should they explode too close. As for stickies You can still push those off points and doors, however, or even push them below the demoman's feet.
You cannot reflect:
Best Answer
Shortly after firing a flare from the Detonator, the Pyro can detonate the flare with the alternate-fire keybind. This creates a small explosion that does damage and lights all enemies hit on fire.
The Pyro can also use the detonation to air-jump. From experience, I'd say it gives about as much distance as a single Force-a'-Nature shot.
The damage stat for "flare detonation" that you see is something you can't quite control. If you manually detonate the flare, the damage is from the "flare detonation" stat, whereas if a flare detonates on impact, it deals the "flare explosion" damage.