What's the difference between Firefox and Mozilla?

Mozilla (Application Suite, also known as SeaMonkey) is a complete suite of web related applications, such as a browser, a mail/news client, a chat client and much more. Firefox is just a browser, which makes it a better choice if you already have a mail client for example. Also, since Firefox is smaller than the whole Mozilla suite, it's faster and easier to use.

Note, though, that Firefox is not the standalone Mozilla browser. The user interface in Firefox differs from Mozilla in many ways. For example, Firefox has customizable toolbars.

How do I spell Firefox? How do I abbreviate it?

Firefox is spelled F-i-r-e-f-o-x - only the first letter capitalized (i.e. not FireFox, not Foxfire, FoxFire or whatever else a number of folk seem to think it to be called.) The preferred abbreviation is "Fx" or "fx".