What are Sass, Less, and Stylus? Why do people use them? How does something like Compass relate to Sass?

They are CSS preprocessors. They are an abstraction layer on top of CSS. They are a special syntax/language that compile down into CSS. They make managing CSS easier, with things like variables and mixins to handle vendor prefixes (among other things). They make doing best practices easier, like concatenating and compressing CSS.
Bonus points for knowing how they differ and/or having experience using them. More bonus points for knowing what things like Compass, Bourbon, LESSHat, Nib, etc are and how they relate.