You can find a simple description of what a feed reader is in this wikipedia article and a good description of the use and technology behind feeds in this Google article.

Basically, a feed reader is a way to aggregate content from various websites in a single place. Instead of having to visit each of your favorite websites every day to see if there's new content, anything those websites publish will appear in FeedBunch. This way you only have to visit FeedBunch to be up to date with all blogs, newspapers, webcomics etc that you follow. You will create your own personalized newspaper and save a lot of time.

For this to work, each website has to make available a special document online that gets updated every time new content is published in the website. This document is called the feed and every new piece of content that is added to it is called an entry. Entries can be news articles in a newspaper's website, new pages in a webcomic, new comments in a blog entry... Most websites nowadays have a feed (or several!).