To cite (a verb, with an associated noun citation) means very much the same, for most students, as 'to quote' (see also the article on referencing). It means 'to refer to, or to call in evidence, a text or witness'. In Law, it can be used of a person, to mean 'called to Court to act as a witness in a case', or 'to name in a document of the court (usually as a defendant)'.
A subsidiary use of 'to cite' in academic work is in describing a quotation that a writer wants to use as evidence - when he or she has only seen the quotation as a quotation, in a separate book. The writer is not quoting the original text from which the quotation comes: she is using a quotation that has been cited in another authority. So a student writer may use a quotation - e.g. 'The law is an ass' from Shakespeare's Much Ado About Nothing - without ever having read that play. The correct way to refer to it is as Shakespeare, Much Ado about Nothing, cited in Wilson, A Hundred Clichéd Quotations.
Beware, by the way: that particular quotation is used by a very foolish character in Shakespeare's play. It is not necessarily to be taken seriously.
For similar-sounding words that are confused, see cite - sight - site.