There are two main font "families:"
Sans-serif, such as Arial, Helvetica and Verdana.
Serif, which includes the Times Roman font you're looking at right now. It's like sans-serif with a little bling added.
The basic typefaces are roman (plain), bold, and italic. Some fonts include such attributes as heavy, light, ultra bold, etc.
Here are some examples:
There you have the basics of typography. Let me know if you have any questions and I'll do my best to help.