When designing a website, the last thing you want to do is pick the wrong font. And by the wrong font I of course mean Times New Roman font.

When picking a font for yourself there are several things to consider:

1) What size will it be
Some fonts squash down into small sizes very well while others do not - try a few fonts out at different sizes and see which ones look good and which ones look bad.

2) Availability
Not everyone may have the font you have chosen on their computer. Using a main stream font such as Verdana or Arial makes sure the user will have it installed.

3) Who uses it?
Find a site you like or a site you want your design to look like and see what font they are using. If it looks good on them it will probobably look good on you.