…that is the question.
There is no one answer.
Footnotes are those little blurbs at the bottom of the page, usually marked with superscript numbers (but sometimes with asterisks and other cool shapes). If you randomly take any three academic books from the library shelf, you’ll find each one of them uses footnotes differently. Some authors uses them solely for citing quotations, some use them to explain archaic words or ideas, and some use them to write a second book within the first.
I think citation is the only use of footnotes which is universally agreed upon. Some people may want you to write the citations in parentheses or as endnotes, but the final result is the same thing. Either way, the little foray to the bottom of the page or the back of the book isn’t going to throw your reader off that much; they can look at them after they’ve finished reading, too.
If you look at a Shakespeare play, you’ll see that half the page is footnotes. This is necessary for modern readers as we don’t teach Elizabethan English classes before we make you read Romeo and Juliet. If you’re ever going to understand what the heck Mercutio is going on about, you’ll need explanations in modern English. We also don’t teach classical literature very much, anymore, so you’ll need the Greek and Roman allusions explained, too. Footnotes are good for these explanations because you’d lose your train of thought entirely if we put all the explanations in parentheses, or if you had to flip to the back of the book. Short footnotes can also be used to show additional sources of information or alternative translations of words.
Unless the goal is to teach you to write without footnotes, most people don’t object to short explanations decorating the bottom of the page.
Now, writing another book on the bottom half of the page… this drives readers right off the deep end. When I see this in a book, I generally put the book back down and walk away. When I see it in an essay, I hand it right back to the student. It’s a sign of inept writing.
If you have a clear thesis and understand the points you’ll be using to support said thesis, there should be no need for extensive footnotes. If every point you make has an additional interpretation or opinion, then you should work that into your essay. If every paragraph has a classical allusion or an archaic word, you need to work these into your essay, too. If the information absolutely cannot be written into the main part of the essay and is not desperately important to the understanding of the thesis, consider using an appendix instead of footnotes; this will be less distracting to the reader.
The term itself should identify the footnote’s position: if it’s taking up anything more than the foot of the page, it’s no longer a footnote. (If in doubt, draw a person on the page; you have up to the ankle to write footnotes.)
Here’s the OWL at Purdue’s page on footnotes, etc.
If you must follow a specific format, MLA and APA formats both discourage the use of footnotes; Chicago recommends them for citation. Make sure you know what’s expected from each format.