If you like moving twice as fast as everyone else, and screaming for an escort as you escape the enemy base with the Flag and the entire enemy team pursuing you, then the Scout is your class. The Sniper is the only class that can kill an enemy on the other side of the map with almost no warning at all. Be careful though, for if the enemy engages you at close range you're weaker than most of the other classes. If you like settling down into a good sniper position and methodically killing every enemy who exits his base with a single shot to the head, then the Sniper is for you. The workhorse in any TFC team, soldiers perform well at any task and specialize in none.
Soldiers are a core class in any offensive or defensive squad.
Carrying more explosive weaponry than any other class, the Demoman clears whole rooms in seconds. He's essential on maps where the use of high explosives can destroy barriers, allowing access to more entrances into the enemy base. His remotely-detonable bombs make him a vicious defender in enclosed spaces. If you like watching the smoke clear and seeing bodies everywhere, the Demoman's your class. His real strength lies in his ability to heal himself and his teammates with his medikit.