Typefaces are important because they determine, on the most fundamental level, how easy it is to read a document. (And after all, all web pages are documents.) At the same time, typefaces determine perception. They should reflect and reinforce the mood of their content.
Most importantly, your font must legible. Some fonts (like the eponymous Bleeding Cowboys) is intended specifically for logos or headings, and would be totally inappropriate for anything longer than eight words. By choosing a font that is easy on the eyes, you improve your site's usability and accessibility.
The typeface that you select needs to accurately reflect the mood of your message. Always consider the audience for which the piece is intended, and then choose a font that achieves your desired perception. For example, a sans-serif modernist font like Helvetica is the font of choice for Apple because it mimics the clean industrial design their customers know and love.