Basically, the Doctor is a Time Lord, an alien being with two hearts who's forgotten more about science and history than you are ever likely to know. He cruises around the universe in his TARDIS a shape-changing spaceship and time machine the size of a small city (it can be bigger on the inside.) however, for budget reasons the TARDIS exterior is frozen in the shape of a 1950's police box and has been for 30 years of broadcast.
The doctor travels with a number of companions, some are aliens, some are robots, a few are from the distant past or future, but mostly they're humans from Earth, England, the present day.
When the Doctor dies, he turns into a new man with all the memories of previous versions. All his likes, dislikes and character quirks change but his core character and motivation "His curiosity, his compassion." remain the same. The Doctor is always an explorer, an adventurer, a myster-solver. He comes from the planet Gallifrey which isn't really there anymore since the new series started, and this makes the Doctor sad.
The sonic screwdriver is sort of an everytool, it can interface with any computer, light candles, open locked doors, pop champagne, screw and unscrew, cut, mend... Basically it can do almost anything except harm. It is not weapon it cannot kill or disable whatever is attacking the Doctor and his friends (except for the occasional robot that is vulnerable to it.)
That should be all you need to know to start any season of Doctor Who.
EDIT: Curse you, Romulan Republic, you forum-ninja!
"
