Doctor Who is one of those TV shows that is so much better than it sounds. I'm not kidding. Would you watch a show that featured a much-older-than-he-looks humanoid alien who occasionally changes his face when he's about to die and runs around the universe with a series of younger companions in a blue police box that's bigger on the inside than it is on the outside? No? Pity, because it's a great show. Tons of other people agree with me, because it's been on (with a couple breaks) for almost fifty years.
The show began back in 1963 with William Hartnell as the first Doctor. Yes, that's what he's called, just "the Doctor." His name isn't actually "Doctor Who," despite what the credits for the first three actors may say. His name isn't "the Doctor," either- we don't know what his name is. "The Doctor" is just fine. And it's accurate, too- because despite the mandates of his species, the Time Lords of the planet Gallifrey, against interfering in the affairs and events of other races, the Doctor doesn't hesitate to step in and meddle when somebody else is trying to mix things up. He makes things better (like a real doctor! Yeah, I know, don't explain the joke...), for which he is rarely thanked, and occasionally loses his life for it.
Yes, he does die. As of right now, he has died (in official canon- there have been a lot of sort-of-canonical and meant-to-be-canonical and is-this-canonical?-no-one's-sure kind of material over the decades since the show first aired) ten times, so he's on his eleventh body. But one of the neat things about Time Lords is that they can do something called regeneration. When a Time Lord is fatally injured (or actually, even if they just feel like a change), they can change their whole body into a brand new one. Of course, they can still be killed-off-for-real if they are put into an environment where even a new body could not survive or killed again before their regeneration cycle is complete, but so far the process has worked well- or at least, well enough.
Right after regeneration, the Doctor is weak, and often has trouble figuring out who he is and what's going on. These post-regeneration brain crazies can make things difficult for the people traveling with him, as only one of these people has been a member of his own species. Occasionally, the Doctor has had trouble remembering who he is or who his companions are. The brain crazies associated with his regenerations from his fourth self to his fifth self (long fall off a radio telescope), fifth self to sixth self (a particularly nasty poison), and from his seventh self to his eighth self (the bullets wouldn't have killed him, but the medical malpractice sure did) were particularly difficult.
With a new body comes an entirely new personality and a completely new wardrobe. This can be a little jarring, and there will always be those fans who piss and moan and scream that the show's been "RUINED FOREVER GARBLARGHL!!!11!!!", but if you give the new guy a chance, he'll knock your socks off. He might even become your new favorite Doctor. He definitely will if you're me- most people have a favorite incarnation, but I love them all. They're all the same guy, really, and I don't care what anyone says, every one of them has something special about him.
Different people have different 'nicknames' for each of the regenerations, but I've always known them by the number of their incarnation (the First Doctor is One, the Second is Two, the Third is Three, and so on and so forth). The incarnations don't actually have different names- on the show, they are all just "the Doctor," but since he's been so many different people, it's helpful to have different ways of referring to each incarnation in informal circles to keep the confusion to a minimum.
Doctor Who is typically divided into Classic and New Who (sometimes called the revival), thanks to those "breaks" I mentioned earlier. The original show was cancelled in 1989, during the Seventh Doctor's run. A TV movie was produced in 1996 that featured the Eighth Doctor (his only on-screen appearance to date), but Doctor Who did not return as a series until 2005 with the Ninth Doctor. If I try to cover all of this in one post, it's going to be painfully long, so I'm going to split talking about Doctor Who up into Classic Who, the TV Movie, and New Who posts. I might not get to them for a while (seriously, there is a lot of material to cover), but I'll get around to it eventually. So bear with me, tiny population of followers!