Web feeds are XML documents summarising the latest updates to a blog or website. These documents are normally viewed through a newsreader, which lets you have a bird's eye view of the latest goings-on at sites whose feeds you've subscribed to. There are two kinds of web feeds: RSS feeds, and Atom feeds. Both are essentially the same, although some software might not support one or the other. If you need a newsreader, Dave Child has compiled a list of newsreader software.
You can subscribe to feeds online if you have an account with Google or Yahoo. In the case of this site, just click on the appropriate button.
Some web browsers, such as Opera and Mozilla Firefox, allow you to subscribe to a feed directly from the browser itself. Firefox does this through what it calls a "Live Bookmark" — if you've noticed the special bookmark with the latest updates from BBC News that comes by default with some installations of Firefox, that's a live bookmark.
Unfortunately, Microsoft Internet Explorer 6 does not support subscription to web feeds (although that's really the least of the issues it has — for webmasters it's a real headache accomodating the quirks of Internet Explorer that other browsers are free of). However, IE7 supports something similar to live bookmarks, so once you upgrade, you should be able to subscribe. (Though to be honest, I'd really prefer it if you considered switching to Mozilla Firefox or Opera.)
A note for old, faithful readers: RSS used to be a supported feed format, but has been dropped in favour of Atom. Your old RSS feed should continue working as expected, but it is highly recommended that you unsubscribe and resubscribe using Atom, as we cannot guarantee continued support for RSS.
If you have any questions about how to subscribe, or if this document is unclear in any way, feel free to contact me.