I like them all, but my three favorites are Engineer, Medic, and Spy. Engineer because of his plain bad-ass-ness. Plus he comes up with awesome nicknames like calling the Medic 'Saw-Bones'. The buildings and taunts are something else I enjoy, because dancing on top of your sentry as it tears apart the advancing payload is very satisfying.
The Medic is cool because he always acts positive, even when what he says is negative. The Meet the Medic video makes him seem exactly like he is: A non-licensed doctor who likes to do macabre things to enemies, heal his friends, and take pride in all of it. Plus, his bonesaw (Or one of the variants) can crit like hell, and I've more than once given a healthy Soldier head (A Solemn Vow to be exact).
Finally, I like the Spy because it's funny when you distract an Engineer from his sapped sentries and dispensers until they blow up, because it means that if you're good, he'll be an idiot and go for you instead of his stuff. He acts like a gentleman (Mentlegen?), and treats his teammates (And occasionally enemies) with respect.
Something I like to say in the voice of the Scout, though, is "If you were from, where I was from, then you'd be from, where I was from!" Which is confusing but makes sense.