• Anime: In the US and other western nations, it is limited to meaning Japanese animation (aka Japanimation, although a lot of people don't like that term). Characteristic features would be large, large eyes, funky hair colors, and often, er, accentuated physiology.... ^_^;; In Japan, anime means animation in general--Warner Brothers and Disney as well as Macross and El Hazard. The modern-day anime style is an adaptation of Disney style (introduced to the Japanese after WWII, by the American occupation), most notably the large eyes.
• Manga: Japanese for comics, in general. Manga make up a huge, huge amount of the published material in Japan (I saw somewhere that it constituted 40% of stuff in print there). Often, popular mangas are turned into anime. Manga, though, goes back far into Japan's past; I'm not really the one to be asked about its history.
• Shoujo: Literally, young girl. When applied to anime/manga--it refers to anime/manga that is targeted towards young girls; one sub-section of shoujo anime is the magical girl variety. Shoujo tends to focus on relationships, emotions, and suchforth. See my shoujo analysis page for a few more details....
• Shounen: Literally, young boy/young man. Shounen anime and manga are targeted towards a male audience. Most mecha shows, martial arts comedies and dramas, and adventure types are shounen, for example.

Note: some shows are rather hard to place in either category. For example, Escaflowne is a mecha show done in shoujo style....