Server-side JavaScript is usually part of an ASP page, and might be used for a number of things, e.g. visit counter, guest book, database lookups, interaction with COM objects or content generation.
Client-side JavaScript is part of what the web server sends to the user's browser, typically a HTML page or a partial ASP page. Normally, this is used to make a web page appear somewhat dynamic. For instance, JavaScript might be used to create data files on the user's computer or to change colors or highlighting based on cursor movement in the browser.
Unfortunately, script language might be used to create viruses as well, and because of this it's not uncommon for users to disable their browser's script engine. Thus, creating web pages that rely on client-side scripting to work is not a good idea.
JavaScript is not related to Java, even though many people seem to think so. I have quite a bit of experience creating JavaScript.