In the last session, I made a Doctor Who joke— which is something I do an inordinate amount of the time— and Gabe expressed his unfamiliarity with the Time Lord, and his reticence to try and get into it, because all the show's history and mythology seemed a little daunting. And I can understand that of course, given that, between its two television incarnations, it has something like 35 seasons. But really it's no more difficult to get into than, say, Star Trek, or Batman, and though the franchise draws on its history, it's really put together in such a way as to be approachable.
So, for no other reason than to make sure that when I make a Dr. Who joke, that everyone can understand it— not necessarily find it funny, because as Mary will tell you, I am congenitally incapable of being actually funny— I thought I'd give a quick two-minute explanation, and some possible starting points. Any other Whovians (hi Jeff!) are welcome to jump in as well of course!
Doctor… Who?
The Doctor (the Who part is not part of his name) is a time-traveling and more-or-less immortal alien who travels through all of space and time in a machine called the TARDIS, which looks like a British police box. He's very old— anywhere from 900 to 2000 years, depending on the story. He comes from a whole race of time-traveling aliens, the Time Lords, who are sworn not to interfere with history and other civilizations. He… he doesn't really do the whole "not-interfere" thing, and is kinda the black sheep of the species. He is also, in the current series, the last of the Time Lords, though this ends up being kinda like Superman being the "Last" Son of Krypton.
The Doctor usually travels with one or more companions, usually female, usually from Earth, usually from modern-day times, so as to provide the audience with a surrogate. And he fights all sorts of alien threats to the universe and to Earth, though he doesn't really "fight" the way Batman fights. He's not, generally speaking, a violent guy, but he'll be merciless if he needs to be— depending on his incarnation.
And this is the other crucial thing about Time Lords— when they "die", they can regenerate. This isn't a case of them regaining 10 hit points per round, but rather of them basically instantly reincarnating. Different face (different actor), different personality, but with the same history. So if someone talks about the Fourth Doctor, or the Tenth, or the Eleventh— they're talking about the fourth, tenth, or eleventh actor to play the role, which can be subtly or vastly different from their predecessors.
And that's really all you need to know in order to jump in almost anywhere in the show, though of course one would do well not to jump in, say, in the middle or conclusion of a big storyline.
Starting Points
If you want to start with the new series— which is definitely more to modern tastes and a little less "creaky"— here are some possible starting points, which should all be available on Netflix.
Blink - Tenth Doctor (David Tennant)
This is one that Jeff recommended, and honestly, it's probably the highlight of the first four seasons of the new show. As Jeff mentioned, it's a pretty "Doctor-lite" episode, so you're seeing the character in glimpses, from the outside. So it might be a good introduction in that sense. It does some neat stuff with time, and the villains, the Weeping Angels, are really great and scary, equalling (if not surpassing) the Doctor's long-time nemeses, the Daleks.
The Eleventh Hour - Eleventh Doctor (Matt Smith)
Another one Jeff suggested yesterday. This is the start of the Matt Smith run (Season 5 of new series), and really, the show is in many ways so different from the first four seasons, that you could easily start here and just keep on watching without missing much of anything. It does a great job of introducing the new companion, Amy, and the Doctor— both in terms of his new personality, and in terms of the basic concept. It also starts a storyline that really extends throughout Seasons 5-7.
Time of Angels & Flesh and Stone - Eleventh Doctor (Matt Smith)
This is the one I suggested, a two-parter. It's less of an explicit introduction episode than The Eleventh Hour, in that it doesn't take pains to explain who the Doctor is, what he does, etc. But I think it's a good introduction in that it's in some ways typical of the show at its best. There's action, adventure, humor, some absolutely terrifying sequences, glimpses of the larger mysteries at the hearts of Smith's run (cracks in time, River Song), etc.
And all of these stories— most of the stories in the whole series, in fact— are pretty self-contained. It's not like you're coming in in the middle of some long soap opera— each episode, or two-parter, or in the case of the old series, each six-parter, involves the Doctor and his friend(s) going somewhere and somewhen else and then getting into some kind of trouble.