Simply put, it's a term used to refer to Japanese animation. Initially, it would appear that anime is no different from the cartoons we were used to watching when we grew up. Yet, after a few episodes, it's obvious that the subject matter, art style, and execution of storylines in many anime differ markedly from the Disney cartoons we have grown so accustomed to.
Why anime?
Gosh, this one's hard to explain. I suppose there exists a stereotype that watching anime is a popular hobby for losers and computer geeks (not entirely untrue, I might add). But personally, though I enjoy watching movies with "real-live" actors, animation provides an entirely new level of surrealism that cannot be achieved with real actors. Fantastic environments can be created without a hitch, costumes never look out of place, and a character's hair always flows just right. Plus, big robots are always neat-o.
Why is anime right for YOU?
Because if you watch anime, it'll make me look better in the eyes of others.