The Eighth Doctor only appears in the 1996 American TV Movie: Doctor Who. He was played by Paul McGann. Eight was kind, enthusiastic, and encouraging. In fact, he was the first Doctor to show romantic interest in a human companion.
The Ninth Doctor was played by Christopher Eccleston during the first season of the revised series in 2005. Nine was in many ways a departure from previous regenerations. Having fought in, and been the winner of, the Time War, he was very much a soldier. Unlike his previous incarnations, Nine carries the guilt of being personally responsible for the extinction of his species -- though he still believed it had been necessary in order to destroy the Daleks once and for all (or so he thought.) Still, despite his harshness, Nine could also be very warm and jovial once you'd won his respect.
I have a soft spot for Nine, as he was my first Doctor. That dorky smile, the big ears, the leather jacket. I wish we could have spent more time with him. He was "fanTAStic!"
And now we get to the Tenth Doctor. Ten was played by David Tennant from 2005-2010 and is many people's favorite Doctor. Though I was stung by Eccleston's departure, Tennet managed to win me over and I now must admit that he is, and probably always will be, MY Doctor. Ten was talkative, enthusiastic, loud, and prone to flights of fancy. He had his darker side though, often quick to anger and vengeance. Occasionally he let his own power get the better of him, as when he tried to change a fixed point in time, declaring that his status as the Last of the Time Lords made him a "Time Lord Victorious," to disastrous results. Besides just liking their company, Ten needed his companions to keep him in check. However, despite his darker side, Ten cared for everyone in the universe, especially his companions, in a way that seemed more human than any other incarnation of The Doctor.