Anime is the Japanese word for animation (taken from the English "animation"); it has worked its way back into the English language as a loanword meaning "Japanese Animation". It is much preferable to "Japanimation" to most fans.
If you've seen Starblazers, Robotech, Battle of the Planets, Speed Racer, or Sailor Moon, you've seen (hacked-up) anime.
In Japan, animation is as important in the motion picture business as live action. The Japanese have developed their own style(s) of animation, and they do not restrict their animation to the family level. Japanese animation titles exist for all ages and cover all movie themes from simple children's videos to hard pornography.
More importantly though, it is relatively consistent in quality: much better than standard American animation (although most do not pass the animation quality of a Disney feature in my opinion). TV series almost always have an ongoing (and developing) plotline, unlike most US shows, and the ideas for both movies and TV are very original.
Anime has gained much popularity recently, and several films have been shown in limited releases to US theatres (Akira, Ghost In The Shell, and My Neighbor Totoro). There is now an anime section in most video stores, and the Sci-Fi Channel now has a Saturday morning anime program at 09:00 ET. In addition, several anime series have been running on TV lately (Sailor Moon and Dragonball Z). But perhaps the most important step forward is Disney's recent acquisition of rights from one of Japan's most successful studios; Disney will be translating these films over the next year or two, and will be showing at least one, Princess Mononoke (Japan's all-time best selling film), in a general theatrical release.
I have started some reviews on the Anime that I own.