The game's theme song, which sounds like a theme song to a badass mystery show. Of course, there's the iconic final eight seconds (referred to as the "ending flourish") which never seems to get less awesome to listen to, especially with its inclusion at the ending sting of every "Meet the..." video.
The background music for "Meet the Medic". "A Little Heart To Heart" and the appropriately titled MEDIC! are now songs in the main menu when you start up the game. The Manniversary update added a gentle clarinet rendition of "A Little Heart To Heart", appropriately named "Archimedes".
Engineer's theme, "More Gun", starts out as a calming guitar melody, but then takes a turn for the awesome when it changes into a dark western themed showdown song. The song's only instrument (for 53 seconds) is a guitar. After said 53 seconds, an electric guitar, a tuba, and a trumpet get added and turns the song into a sinister-but-awesome western song. It's as awesome as it sounds.
Right Behind You. This theme has been known to induce panic and horror if played in-game. It's chilling, quick-paced, and gets the message across that the class it represents (Spy) is not to be underestimated.
Drunken Pipebomb for the black Scottish cyclops. It deserves credit for actually making bagpipes sound awesome.
Scout's theme, Faster Than a Speeding Bullet, a fast-paced combination of a catchy piano riff and an almost inhumanly fast bongo solo. In can show you he's quite powerful, if you know how to use him.
The Heavy's jazzy theme. Short and simple, yet catchy. It encompasses the vibe of the entire game in general (black comedy in a '60s setting) so well that it's known simply as the 'Team Fortress 2 Theme'.
The Soldier has "The Art of War", an incredibly badass drum song with periodic brass.
Pyro has the quite unnerving Dreams of Cruelty. Other than "Do You Believe in Magic", that is. And that little child's laugh at 0:47...
"ROBOTS!" An extremely catchy, rapidly-escalating theme that, with its off-key notes and high, whining, unnerving sound makes it distinctly inhuman. It just keeps getting bigger and bigger and more and more insurmountable - this is the sound of a horde of robots, swarming you, more than you can ever count. "The Calm" is an unbelievably epic buildup to the above tune.
This muzak-esque tune that was intended to play at the upgrade stations. It's so catchy that some people have theorized that it was removed because players would just stand around and listen to it instead of fighting.