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Facebook applications are small programs that run inside Facebook. They're similar to web browser plug-ins (like video players) in that they let you do a little something extra that you couldn't do before you installed them.
If you've had a chance to put Facebook through its paces, you're probably already familiar with the site's built-in applications, which include Groups, Events, and Photos. But people who aren't on the Facebook design team—folks known as "third-party developers"—have written about a bazillion others, such as iLike (which lets you add music clips to your profile) and Weekly Schedule (which lets you coordinate a graphic version of your schedule with your coworkers' or friends').