Okay, there are a lot of definitions for this word, and most of them are crap. I'm going to give you the complete, unabridged, truth:
Definition 1: Anime is the French (yes, French) word for animation. Simple, no?
Definition 2: Anime is a word used in Japan for any animation. It is taken from the french word.
Definition 3: This is the definition that is most commonly used and encountered in the United States. It means 'animation made in Japan, or by Japanese people'. That's it.
So, there are three definitions. I hope we're all clear now.
French guy: Ze anime is good, yes?
Japanese Guy: The first anime, Gertie the Dinosaur, was made in 1912.
American Guy: Anime is animation from Japan.
Three different definitions, got it!?!?