A biography is an extended piece of non-fiction that tells the whole story of a person's life. Unlike an autobiography, biography is not written by the subject of the book.
For a hugely famous biography, check out James Boswell's epic tome Life of Johnson, which is all about one of English literature's most towering figures, Dr. Samuel Johnson. It is widely considered a paragon of the genre, although Boswell may have made up a few things. Hey, it happens.