Proper citation of sources is absolutely vital to essay writing. You must acknowledge your sources for a variety of reasons. Foremost is that of credit: it is important that you give proper credit to the work of the various scholars whom you have employed. If you don't, then your paper is open to the charge of plagiarism. Plagiarism is a form of scholarly theft that occurs not only if you copy someone else's words and pass them off as your own, but also if you use someone else's ideas (even if you use your own words) without acknowledging them properly. Plagiarism isn't limited to deliberate fraud; it is something that can happen simply through inexperience in how to use sources. This is really important to remember: plagiarism can happen unintentionally!
When you do draw on someone else's arguments in your paper, either use your own words to express it (you still need a footnote) or use quotation marks (again, a footnote). If you employ someone else's words, but don't use quotation marks (say, because you changed one or two little words), then even though you cite them properly in a footnote (as you must), you're still being fraudulent in passing off their style as though it was your own. So when I say use your own words, that really means use your own words, it doesn't mean just change a couple of vocabulary items. I know that using the words of the original authors often seems to make sense, since they're the ones who had the ideas and expressed them so perfectly; but if that's the case, then just go ahead and use their words, in quotation marks.
A note: sometimes the need to avoid plagiarism, and the lack of confidence in one's own ability to summarize or express arguments can lead students to producing papers that consist of nothing but a string of quotations. This is obviously not good, either. Your paper should express your own responses to the arguments and issues. Think about what various authors are saying; don't just report their words.
Another important reason for citing one's sources is to enable the reader to go directly to the original source of the information or the idea. That way the reader can learn more about the author you have cited, clarify any misunderstandings, etc. This is one reason why the fluidity of electronic sources can be problematic. A Web site that's there one day and gone the next isn't particularly reliable as a source of information. (See further below on proper citation of electronic sources.)
Furthermore, since the essence of most essays is to provide analysis of an issue that is probably in some respects controversial, you need to distinguish between scholars who have produced conflicting interpretations (to learn more, visit the Analysis section).
In Classical Studies specifically, it's very important to acknowledge not only the works of modern scholars throughout your paper, but also the material that's been drawn from the ancient sources. The more you study the ancient world, the more you'll realize that the quality of our primary evidence varies enormously. It makes a vast difference whether a particular snippet of information comes from Thucydides (pretty trustworthy) or from Justin (pretty nuts). You have to let your reader know the original source.
Proper acknowledgement of the work of other scholars can only be done through some kind of footnoting system throughout the course of the paper. Simply listing a work in your bibliography isn't enough (unless it was only useful to you in a general kind of way). Obviously, when you quote something directly, you know that you have to enclose it in quotation marks and footnote the reference. But I repeat that you must also footnote ideas, arguments and specific information. On the other hand, widely known and uncontroversial facts don't have to be referenced. You don't need to provide a footnote to support the statement the Peloponnesian War began in 431 BC. But you should provide a footnote to support the following statement, which is a matter of interpretation (in this case, the view held by the ancient author): the Peloponnesian War began as a result of Spartan fear of Athenian imperialism (footnote: Thucydides1.23). The reason for this is that this is only Thucydides's opinion; it may be a valid one, but it's one that remains open to question.
In Classical Studies, primary sources (chiefly ancient authors) should be referred to by their traditional divisions, e.g. Herodotos 8.45 (= Herodotos, Book 8, chapter 45). Since Herodotos only wrote the one work, it does not need to be referred to as The Histories (as it is labelled in many translations). Unless it is absolutely necessary, ancient authors should not be referred to by the page number of a modern translation (if that is absolutely the only way to do it, then you must ensure that all the publishing information for that particular translation is also included with the essay). Once again, this is a simple courtesy to the reader, whose translation of a particular author may not be the same as yours, or who may even be looking at this author in the original Greek or Latin.
Citations of ancient authors, because they're so brief (Herodotos 8.45, Thucydides 1.23, etc.) are usually best enclosed in brackets within the text of the essay, rather than tucked away at the end in an endnote or footnote.
There are a variety of acceptable formats for proper referencing of secondary sources. Any style manual will lay them out (see also the section on Style). Here are some of the most vital requirements:
Minimum (and often maximum) required information: author (and perhaps work) and page number. Since all works consulted must be listed separately in your bibliography (with a complete citation), your note should only have to read something like this: Syme p. 56. If you have more than one book by Syme in your bibliography, then you could specify that you are referring to The Roman Revolution as follows: Syme Revolution p. 56 or you can specify by date (Syme 1962 p. 56) .
Once you adopt a particular format, stick with it. It does not matter (to me) whether you use footnotes or endnotes or enclose your citations in brackets within the text, so long as you do the same thing all the time. (While I am not overly particular about which style you choose, others may be; check with your own professor if you have any concerns.)
Make sure your reference communicates what you wish it to. This is often best done by keeping the format as brief as possible.
Electronic sources need to be cited, just as more traditional media do. There are a couple of particular problems with electronic materials. For one thing, as I pointed out elsewhere, Web pages are not peer-reviewed in the same way traditional publications are. Anyone can put up anything, and very frequently there's not even any way of telling who put it up. A general rule of thumb you could follow (and this would relate to use of encyclopedias as well): if you can't find an individual's name attached to something, then it shouldn't be part of your research. Furthermore, the fluid nature of the electronic medium means that a source can be very ephemeral. So use this material with these cautions in mind, and when it comes to citation, then use a style manual for electronic sources. See the UW library site for more help and all forms of citation.