When the Doctor 'dies' he regenerates as a new Doctor, a new version of him. He's not exactly immortal, as one day he will run out of regenerations, but he will be around for a long time.
This is sometimes also known as the actor playing the Doctor wanting to move onto new things.
It gets a bit complicated, as the series has been running for decades, and there have been lots of spin offs like books, radio and audio dramas, which leads to debates about what is and isn't cannon. However, the beginners' explanation is the Doctor you are currently watching is the 9th regeneration of the Doctor (Christopher Eccleston is the 9th actor to play the Doctor on TV).
The 9th Doctor is also a bit of a comeback special, as the series had been dropped for years by the BBC, but they finally bought it back, and the nation was happy again
