Manga
Manga is Japanese comic books. It is an extremely popular medium in Japan and is a multi-billion dollar per year industry. Unlike in America, comics are just read by super hero loving teenage boys, they are popular among people of all ages and their topics span from action to romance to business.
Anime
The word anime is an abbreviation of "animation." In Japan the word anime is used for all animation, but in America refers to a specific style of animation that originates in Japan.
Anime has become popular in America and many TV shows and movies are translated to English for US audiences. Dragon Ball, Naruto, Pokemon, Sailor Moon, and Detective Conan (Case Closed) are some TV shows popular in America.