A quick search turned up this similar question on another site.
In short, quotes within quotes are the main usage of enclosing single quotes. Another common usage is for quotes in a title or headline. Of course, the same character is used for apostrophes when typing, as the QWERTY keyboard has never had separate keys for an apostrophe and a single quote (the look of which CAN differ subtly depending on the typeface).
According to this other page, in British English, single-quotes and double-quotes have historically been reversed in terms of preference; normally, quotations should be enclosed in single quotes, and double-quotes are used for internal quotations. This preference is still present but fading in the UK, and American usage has always preferred double-quotes for the initial quotation and single for internal quotes.
Unofficially, I have sometimes seen single quotes used to denote thought instead of speech. More commonly, the thought is italicized, or no formatting is applied at all; these styles are more correct, and if you had written a manuscript employing single quotes in this manner, they'd likely be removed or replaced by your editor or typesetter.