I saw this discussed in the context of tags on meta, and surprisingly, there doesn't seem to be any authoritative-looking answer when I Googled aside from some forum threads which I don't know if I should trust.
Anime and Manga are two different storytelling media. They both originate in Japan, and are closely related, but are ultimately two different things. The confusion between the two arises mostly because it's often the case that the same story will have both an anime and a manga version. The terminology will vary a little bit depending on whether the person you talk to is a Japanese person or a westerner; I'll try to point out where this happens.
Anime (アニメ, a shortened form of アニメーション, which is literally "animation" when written as a loanword in Japanese) are Japanese animated cartoon videos. These air on television or are released to home video. Producing an anime is a large undertaking, and requires the work of an animation studio with a large number of people.
There is some debate as to whether non-Japanese cartoons qualify as anime. A Japanese person would say that any cartoons at all can be included as anime, including western series like Avatar: The Last Airbender or Spongebob Squarepants. Most people outside Japan use the term solely to refer to Japanese-origin series, or at least those which are inspired significantly by Japanese anime (so Avatar might count, but Spongebob certainly wouldn't). For more information, this Anime SE question might be helpful.
An image from the Saint Seiya anime
Manga (漫画, which could be literally read as "whimsical drawings") are Japanese comics. Unlike anime, they're typically black and white. Manga are often used as the basis for anime, but not every anime is from a manga and most manga are never made into anime. Manga usually only require a small number of people to produce, at minimum a mangaka (who is the author, illustrator, and all other major roles) and an editor. Unlike western comic books, most manga are read right-to-left.
As with anime, Japanese fans wouldn't have much problem labeling comics from other countries as manga. In the English-speaking world, it's more complicated. OEL Manga (Original English Language Manga) is now a standard term for comics like Megatokyo which are inspired by manga but produced in English-speaking countries. There's also manhwa (Korean origin comics) and manhua (Chinese origin comics), both of which heavily borrow from manga. Japanese people would usually label all of these as Manga, but English-speakers will usually make the distinction.
A couple of panels from the Saint Seiya manga
Manga are comic like books. Animes on the other hand are animated films. What people would call a cartoon (although im pretty sure there's a difference and some people would kill me for calling an anime a cartoon).
Many good selling Mangas are made in Animes - that is, animated TV series. For example, the Naruto animes are based on the Manga with the same name.
One is a comic version, the other is an animated version. Usually the manga or anime have a book of it and a show. Both were made by the Japanese but they have a lot of differences besides the fact that one is a show and one is a book. Anime is animation of a cartoonish show and manga is book of pictures or comics (also graphic novels). They are not the same!