In short, it's a british science fiction TV show. But it's more than that, it's actually the longest running science fiction TV show, ever. Including Star Trek and Stargate, which are close seconds. The first episodes were aired in 1963 and get this: most of the early episodes are lost forever due to the fact that the BBC used to have a policy of reusing tapes and taping over old shows. They're desperately trying to contact third parties to recover as many episodes as possible.
Anyway, the reason you hear about it recently is that, after a long hiatus (for a long time, the last appearance of the Doctor was the 1996 TV movie) it's been revived in 2005, and has been enjoying quite a lot of success, enough to guarantee a regular season every year since.
They manage to keep the show alive by using a clever gimmick: the Doctor never dies, he regenerates. So every now and then he's played by a different actor. We're now at his Eleventh incarnation, played by Matt Smith.
This also means that it has evolved significantly throughout the years, and old episodes look and feel very different from the new ones.
One thing: "Who" is not his name. He goes by "The Doctor". Also, it's always "Doctor Who", never "Dr. Who".