There are three basic categories of fonts: Serif, Sans-Serif and Script.
Serif – A serif font is one that has serifs or the extra tails on the end of each letter. The most popular serif font is Times Roman, others include Bookman, Century, Garamond, Lucida and Palatino. The list goes on.
Sans-Serif – A sans-serif font does not the serifs or extra pieces at the ends of the letters. The most popular sans-serif font is Arial, others include Century Gothic, Helvetica, Lucida Sans, Tahoma and Verdana.
Script – A script font is one that tries to emulate handwriting. Some script fonts are Brush Script (ugh), Edwardian Script, Freestyle Script, French Script, Papyrus and Vivaldi.
There's LOTS more to say on this, but this is generally what you're after.