Using Google Web Fonts
Google Web Fonts is a great resource for custom web fonts. It is free and has a huge library of fonts (508 different fonts at time of writing).
A lot of people are saying that while the library is large, the quality of most of the fonts is not that great. While this is true, you just have to find the high-quality fonts that do exist. There are a couple of useful sites on the web that help you choose fonts out of the huge selection. Beautiful Web Type is a great resource as is Google Webfonts that Don’t Suck. Alternatively, just play around with the various fonts (there is a handy chrome plugin).
It is really easy to include whatever fonts you want in you site, just beware about having too many, as it does increase your page size as well as not looking too good if you have tons of different fonts that don’t complement each other. A good guide for choosing fonts is A Beginner’s Guide to Pairing Fonts.
I’ve recently changed what fonts this site uses, using Google Web Fonts; I went with Oswald for headings, and Lora for the body copy.