Back in the day, if you liked a site, and you wanted to make sure you saw all their articles, you would add it to your Bookmarks, or maybe even make it your Home page in your browser, and check back regularly. Today, you would just add it to your RSS reader instead. (In fact, most web browsers will even double as RSS readers for you.)
RSS (Really Simple Syndication) is part of almost every major website. You'll see it denoted with the icon The idea is like a magazine subscription. Instead of going to the newstand (website) to get your content, the content is sent to your mailbox (RSS reader) whenever it is updated.
This is this handy for the obvious places, like news websites like CNN or the New York Times. It's fantastic for things like friends' blogs, which don't necessarily get updated all the time. And, once you start using it, you'll notice that it's integrated into lots of sites. Flickr lets you add your friends' photo streams, so you'll get new entries every time they post pictures. Farecast (aka Bing Travel, apparently) lets you use it to track airline fares. Google will update your feed every time it scans a page with your name on it (or any other piece of text you're tracking.) Can't remember when your favorite webcomic updates? You can get a feed. And, of course, Pobox will give you a feed of all your spam (or just the results of one of your views, if you prefer.)
It's easy to get started, too! There are desktop programs like Feed Demon and NetNewsWire. I prefer web-based programs like Google Reader or Bloglines, because then they're up-to-date on any computer (or iPhone) I use. Most programs will let you paste in the URL of a site you like, and then they'll scan the site for feeds. Other sites will just present buttons that automatically add their feeds to common feed readers.
It seems at first, that you're just replacing one website with another. But as you add more and more feeds into your reader, and you stop losing track of blogs, pictures, news and sites that you enjoy, you'll wonder how you ever lived without RSS. With RSS, rather than chasing content across the Internet, let it come to you.