http://www.christiananime.net/terms.php
Anime - Japanese cartoons, so to speak. They are animated, usually 30 minutes in length and episodic (meaning they're like American television shows), and range from G-rated Children shows to basically animated porn
Manga - Commonly compared to American comic books, manga is the print form of anime. In fact, many anime originate from a successful manga run.
It's commonly said that it originated between the end of World War 2 and the 1960's, when people who made movies couldn't afford the physical costs of making sets and transporting people (this is what I heard, correct me if I'm wrong!), and took it's cues from American cartoons, but developed into it's own style.
I personally prefer manga, since the storyline tends to be longer/more involved than in anime, but there are many anime that I like just as much as their manga counterparts.