Japanese Animation. The term technically began its life as the English word “animation,” which Walt Disney Studios probably copyrighted around the time they were making Donald Duck cartoon war propaganda.1
When the Japanese were making their own cartoon war propaganda,2 they needed a way to describe the process, so they borrowed the word “animation” from us in the same way we borrowed the word “karaoke” from them, which is to say it was butchered almost beyond recognition (much like the act of karaoke itself, actually). It was later shortened to “anime” and used as a blanket term to describe all animated projects.
Eventually English speakers picked up the term, this time using it to refer specifically to Japanese animation. These days the linguistic lines are growing fuzzy again, as American animators are beginning to create works influenced by the Japanese animation style (such as the excellent Avatar series), and fans can’t agree on whether this counts as “anime” or not. One of these days someone will probably come up with a new term to describe this sub(-sub?)-genre, and a whole new group of nerds will be created and subsequently mocked. And thus the circle of life continues.