The Doctor is an alien, a real one. Not from the physical point of view but from the mental sides of it. He is inhuman, fearless, funny, reckless... ''a mad man with a box''! Mad men, we like! Within a single episode, you can laugh, cry, be scares, exited... and sometimes all of those at once.
The Strength of this TV show isn't in the script, well the storyline is very good overall, but it is in the characters that all fans can't get enough of. That also includes the Doctors companions, who ultimately turn out to be pretty awesome after a few trips with the doctor. The show has been going on for a very long time, and the well doesn't look like its about to dry up. Also Torchwood, as spin off series to the show tends to prove that it's not gonna dry up soon either. Advice : listen to the seasons in order, do not skip parts!