Anime: Japanese for "animation" or "cartoons." In American fandom however, it's used to refer to any form of animation produced in Japan. Anime comes in the form of animated TV shows or movies, video games, and comic books.
AMV: stands for Anime Music Video. AMVs are videos made by fans that feature scenes from anime together with audio from another source such as English songs, dialogue or music from another TV show, or movie trailers. The best AMVs are praised for their good video editing techniques and cleverness in choosing the best anime scenes to go with the chosen song or dialogue.