What is a Style Manual?
A handbook or guide that illustrates the accepted format for citing your sources (references) in term papers, theses, articles, etc.
Why should you cite your sources?
Well, mainly, because your professor expects you to (so you won't be accused of plagiarism). That said, citing your sources also shows that you have done research using authoritative sources. Also, bibliographies act as further reading lists and help guide the reader of your paper to additional sources on that particular topic.
Which style should you use?
In many cases, your professor will tell you which style they prefer. If they don't, here's a general break down:
APA: Business, Education, Psychology, and other Social Sciences
Chicago (Documentary-Note Style): Literature, History, and the Arts
Chicago (Author-Date System): Physical, Natural, and Social Sciences
MLA: Literature, Arts and the Humanities
What is a Bibliography?
A list of sources (books, articles, etc.) used or considered by an author in preparing a particular work. Depending on the Style Manual you are using, the bibliography might also be called the "reference list" or "works cited"
(Adapted from the Arkansas State Univ. Library Citation Guide)