- The Doctor is the last Gallifreyan Time Lord. He is always on the move, always traveling through time and space. He is compassionate (especially toward humans) and good and incredibly clever; usually he's the smartest guy in the room. Gallifreyans have two hearts, though otherwise they appear humanoid.
- The Doctor is the same character, no matter who is playing him. Though, accordingly, his mannerisms, temperament, preferences, and a few other things change with his different form. It's really kind of cool. The reasoning behind the different forms is that the Doctor can regenerate instead of dying, which means he is very, very old. Oh, and his name is not "Doctor Who". The title of the show is a sort of smile & nod at the frequently asked question when he introduces himself as "The Doctor".
- There have been (to date) 11 actors who have played The Doctor. So, when people refer to David Tennant as the tenth Doctor, they are counting all the actors, way back to when the classic show originated in 1963.
- The Doctor likes to have a companion (or two) with him. The length of their time with him varies. He is not romantically involved with them, though many of them are rather fond of him, and it's implied that he may have fallen in love with Rose.
- The Doctor prefers to win without killing.
- The Tardis is a blue police phone box, but that's just its form. It's really a time-machine-space-ship that's "bigger on the inside". It has some pretty cool powers and is connected to the Doctor's sonic screwdriver, which he uses to open things, detect things, and sometimes it even helps defend him and other people. The Tardis makes those who ride in it able to understand any language.
But if you want to dabble your toes in the water first, you might start with a few fan favorite episodes, which stand alone quite well. "Blink" and "The Girl in the Fireplace" were both written by the brilliant Steven Moffat, and "Vincent and the Doctor" is a very good entry episode as well.