A server-side language is one whose code is meant to run on the server which holds the page content; the opposite is a client-side language, which is run on the computer of the user viewing the content.
The main advantage of server-side languages is that they are almost always more powerful. Why? Mostly because they're not relying on the capabilities of an unknown system that could be anything from a 286 to a dual P4. More specifically, it doesn't rely on the interpretation of that language by the browser; because it will always run on the server, the programmer can test it and know exactly what it will do. Common examples of server-side languages are PHP and Perl.
Understanding the above is the easiest way for a user to know which is which. If you don't use any features of the language which give it away, they could look at the file extension (e.g. answerbag.com/a_edit.php is PHP and therefore server-side) or page source (client-side code is part of the page and will be visible in the source; server-side code isn't and won't).