I've recently played Diablo 1 and 2 and I'm totally hooked. Now I've been wondering what genre this game belongs to, so it would help me find similar games.
The genre has a few popular names, including:
And sometimes even the term "roguelike" is used - there is a vast difference between the roguelike genre and games like Diablo, but the latter are obviously influenced by the former.
The most common term I've encountered is "hack and slash". There are many, many games that belong to this genre, especially since the success of Diablo.
The excellent Moby Games site has a page about the genre, which briefly outlines what defines it as well as providing a list of all the games it considers to be "Diablo variants".