I really like the Second Doctor. Unlike Hartnell, Patrick Troughton actually seems like the central hero of his own show, rather than just somebody's grandfather. He's still old, but he's considerably more spry. He's a bit of a clown without actually seeming like a fool, which eventually became a trait that pretty much all the doctors embody to a greater or lesser degree. He's also genuinely funny at times, as Troughton proves to be a great comic actor.
Another thing the Second Doctor has going for him is the most adorable pair of companions ever created, Jamie McCrimmon and Zoe Heriot. I mean, Victoria was okay, but Zoe really proved to be the missing variable in a formula of concentrated awesome. You have the Doctor, a funny little man who travels time and space, so who should he invite along? A bright young 18th Century Scotsman in a kilt, and a petite science prodigy from the future in a sparkly catsuit. I mean, obviously.
Unfortunately, a lot of Troughton's episodes are lost, and a lot of the ones that survive are pretty creaky and dated. Also, he spent a lot of his time fighting the Cybermen, who I have to admit are my least favorite of the major Who villains. Still, though, the scene in "Tomb of the Cybermen" where he talks to Victoria about his age and remembering his family is one of my favorite Doctor moments ever.