So far there have been 11 Doctors. This may stay the same for a while as the current Doctor, Matt Smith, has lately said that he is in no hurry to leave, and he is very popular so the BBC isn't likely to change him for a while either.
The first Doctor was played by William Hartnell in 1963. He stayed for three years, which is a fairly average run, and followed by Patrick Troughton, then Jon Pertwee who kept the role from 1970 to 1974 and was the most popular up to then - many who remember him still feel he was 'the best Doctor'.

After that, the fourth was Tom Baker, who if anything was even more loved than Jon Pertwee - he was famous for always wearing a very long scarf and a hat, and he played the Doctor for seven years.

More recent Doctors have included Christopher Eccleston, with Billie Piper as his companion Rose, then David Tennant and now Matt Smith - in the news now, as a new series is about to begin. You can find a full list of actors here.