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I notice in some of the books I read that the author sometimes leaves explanatory notes that refer the reader to a book but the note is not written in a citation style.
Do footnotes and endnotes have to be written in citation styles such as MLA or Chicago? Or are citations only for bibliographies?
As Oxinabox says, if you are writing for a journal, they will almost always, if not literally always, have standards for footnotes and endnotes. Usually they'll say to follow MLA or APA or whatever style guide. Some may have their own rules.
If you are writing a scholarly book, the publisher may have a specific style guide. If not, or if you are self-publishing, you should still at least follow the spirit of the popular style guides if you want your work to be taken seriously. A scholarly book with no footnotes, or with footnotes that don't enable a reader to find the original source, is likely to be dismissed as "not a scholarly book".
If you're writing a non-fiction book for general readers, i.e. not for professionals in the field but for ordinary people, you have a lot more leeway. If the authority of the sources you are quoting is important, than you should have decent footnotes or endnotes.
Many non-fiction books rely primarily on "common knowledge", well-documented facts that don't particularly require foot notes. Books that are intended to be summarizes of generally accepted information, like introductory books, often don't have footnotes. If a book is primarily analysis, commentary, or opinion, it may not need footnotes.
Works of fiction rarely have footnotes at all. Normally you don't want to interrupt the flow of a story with a footnote. Fiction translated from another language may have footnotes to explain difficulties in the translation. Books written in another culture may have footnotes to explain cultural things that the author assumed but that may not be clear to readers in another culture. Etc. But these are relatively rare.
The various citation styles such as MLA and Chicago, often have a footnote style, or a defacto one -- often the same as the bibliography style.
Some citation styles are by normally written as an endnote -- within the bibliography -- Eg IEEE.
It very much depends on the context as to what is appropriate. In formal academic writing, eg a journal article there will be a definite and concrete answer. Almost certainly you can look it up on the journals website. Similarly for writing a textbook.
If it is in an academic context, for a field that uses LaTeX, you might be interested in the bibLaTeX \footcite macro. There is some explanation of it in this TEX.SE question.
In less formal situation, eg an expository nonfiction book, there will likely be guidelines, you should speak to the editor.
In a novel, I would be surprised if there were guidelines, beyond perhaps the normal referring to media works. Its mostly something to be handled in late stage when working though with an Editor/Copy Editor. Just leave enough information to handle it then. Eg Author, Book, Chapter, Edition.
