
First, you'll often see the terms RSS (really simple syndication), ATOM, news feed, web feed, or syndication used interchangeably (which is not entirely correct, but close enough for this purpose). Feeds are used by some websites as a means of publishing their latest content. Blogs use feeds to publish their latest posts, most news sites use it to publish the latest news stories. My feed will publish my latest paintings and newsletters. 'Publish' means that the website provides a list of its newest content to any software that wants to use it. These feeds are intended to be read by software, not read directly by people. That's the boring part...
The interesting part is in the software that can
make use of these feeds. The generic term is a 'feed
reader' - they give you a single consolidated
view of the latest items from your favourite websites. You'll have one
page where you can see what's new, so you don't waste time checking
individual sites and trying to figure out what's new - the software
figures that out for you. Say you like to watch the CNN and BBC
websites for current news and you watch
a few other blogs too because they're interesting. Add all of these
individual feeds to your feed reader and you'll save yourself a lot of
time because you'll see at a glance which sites have something new since
you last checked.
Exactly how you subscribe to a feed depends on what
feed reader you're using, but often it's just a matter of clicking on the

icon when you see it on the website you want to subscribe to.
(Sometimes the icon will say RSS, ATOM or XML.)
There are a lot of different feed readers you can get for free, but here are a few of the most popular: