The Doctor is the last Time Lord of planet Gallifrey.  He is 900 years old and now travels through time and space, battling his ancient enemies like the Daleks, the Autons and Cybermen with the help of a female partner and, sometimes, a robot dog!  The Doctor travels through time and space via his TARDIS (Time And Relative Dimension/s In Space) machine, which is bigger on the inside than on the outside.  His trusty sonic screwdriver has saved him from tight spots more than once, being capable of opening any lock, be it mechanical or electronic, except deadlock seals.

THE TIME LORDS
The Time Lords are a race of people who were so powerful because of their ability to travel through time, they simply watched the universe go by.  They reside on Gallifrey which was described vividly by The Doctor in Gridlock, series 3.  Unfortunately, a race called the Daleks (from planet Skaro) initiated a Time War, the last great Time War, which resulted in the destruction of all the Time Lords except The Doctor, and almost all of the Daleks (almost).  Ever since then, The Doctor has been travelling in his TARDIS (the only TARDIS that remains), occasionally accompanied by a female companion or, in one case, a robot dog called K9.

Extra info - Time Lords have two hearts and can regenerate themselves when close to dying, although they can choose not to regenerate if they wish (see Last of the Time Lords, series 3)

THE TARDIS
The TARDIS is a living machine which is linked telepathically to its owner (ie. The Doctor).  It can help with the regeneration process and used to be able to blend in with the time period before its chameleon circuit was cut.  The TARDIS is powered by a black hole called the Eye of Harmony, created when a Time Lord named Omega sacrificed himself in the making.  The TARDIS is virtually indestructible but must never be materialised inside itself.

The Doctor's TARDIS is currently stuck as a police phone booth, which were common around the 19th century and used specifically for calling the police.  It's inside can also change to suit the current Doctor.  In the past, it has contained a swimming pool, a library and even a garden.  The TARDIS seems to have infinite space in its interior.

NOTE: Always carry a spare vial of mercury for the TARDIS's fluid leaks.

THE DOCTOR'S COMPANIONS
The Doctor's companions have varied much, almost as much as he regenerates.  His companions always seem to be female, and they help him throughout his journey through time and space.  Although The Doctor keeps his feelings to himself, he fell in love with his more recent companion, Rose Tyler (Series 1 and 2), whom he saved once by kissing her to draw out the deadly power of the Time Vortex she had absorbed into himself, causing him to regenerate into the 10th Doctor (David Tennant).

In Series 3, The Doctor's companion is called Martha Jones, a doctor (a medical doctor) in training.  Martha first met The Doctor as John Smith, who admitted himself into the hospital where Martha worked because of strange occurences that day.  Once everything is back to normal, Martha spots The Doctor heading towards his TARDIS and watching her.  She turns around for a second and turns back to find that it has disappeared!  Later that night, after a party, she sees The Doctor again, observing from the corner of an alleyway.  He heads down it and Martha follows him to the TARDIS.  After demonstrating the truth of its time-travelling capabilities, she joins The Doctor, and he promises Martha one trip for saving him at the hospital.

THE DOCTOR
The Doctor is the last of the Time Lords and has lived for 900 years.  He has no real purpose in life as such, so he travels around on impulse and always seems to find trouble somewhere, which seems more common if he has a partner with him.  Otherwise, he visits past events and helps in the creation (helps Benjamin Franklin discover static electricity) or simply watches those in progress.

The Doctor is summed up very nicely by Timothy in The Family of Blood, series 3: "He's...like fire and ice and rage...He's...like the night...and the storm and the heart of the sun, he's ancient and forever...he burns at the centre of time and he can see the turn of the universe.  And...he's wonderful!"

THE SONIC SCREWDRIVER
The sonic screwdriver was first invented by the Doctor after his first regeneration (ie. the second Doctor).  It has many purposes ranging from opening all locks mechanical and electronic excepting deadlock seals to acting as a sort of scanner to determine a person's internal health/state (as seen in 42, Series 3).  It can also operate parts of the TARDIS from a distance (one example is in The Parting of the Ways, Series 2).

One function that it cannot perform, is the ability to kill another life form.  The laser screwdriver (invented by the Master) has this function and he uses it in The Last of the Time Lords, Series 3.  The laser screwdriver seems to have no other purpose and it has not been seen to open any doors or perform any function similar to the sonic screwdriver.

Despite it's inability to injure, incapacitate or maim another, the Doctor has tried to use it to threaten others such as in World War Three where he threatens to "triplicate the flammability" of some alcohol, thus destroying himself and some Slitheens.  Unfortunately, it wasn't threatening enough as the Slitheen perceive he is lying and the Doctor is forced to rethink his strategy.

The Doctor has been occasionally forced to rebuild his sonic screwdriver, often creating something similar to it but they are often given a different nickname such as the "sonic suitcase".  Once, the Doctor's screwdriver was burnt out during an attempt to increase the radiation level of an X-ray by 5000%.  He rebuilt it in his TARDIS and takes Martha with him in order to "road-test" it (see Smith and Jones, Series 3).