Anime is Japanese animation.
To elaborate on your other questions, it can be as weird as people make it out to be, sometimes it can be even weirder, and sometimes it's not that weird. As for why geeks love it so much, one theory I have is that many more popular series have quite a lot in common with superhero comics; people seem to like stories about super people fighting super-villains, whether they were always super or normal people turned that way. Also, there seems to be a substantial overlap between video game and anime fans and cultures (look at this, or any, gaming forum) for whatever reason.
As for some recommendations, I'll just give the boilerplate of Miyazaki films and Kino's Journey. If you're willing to go into an anime that, while not a shonen-fest, is more "anime-y," however, try Darker Than Black.