Tags, in short, are keywords attached to objects to assist searching and browsing. Most important for images and videos, tags allow computer-users to find items which cannot be indexed by standard text-based means.

What does that mean?
If you were to look for a document containing your last family vacation's itinerary, you would search for "itinerary" on your computer, and easily find it, since the file contains that word on the top of the document. But searching for a video of that vacation would prove to be much more difficult. Tags allow you to mark that video with the words "family vacation 2006 Hawaii". You could then search for any or all of those words, and viola, you've found your video.

Why use tags for text?
Although essential for media, tags are useful for text items as well. Depending on how you use them, tags can help you find the right information quickly.
Tags can be used sparingly as category names, placing chunks of data into multiple categories easily. A searcher can add more tags to his search to drill-down to the most relevant data.
Tags can also be used to highlight certain words that may be contained within a block to easy searching. For example, an article might have the words "family politics" in it, but you only want the article to appear when the word "family" is searched for, to distinguish it from your "politics" articles.

For more information about tags, see the explanations on these other sites:
YouTube
Yahoo 360
Flickr
del.icio.us
Wikipedia
Technorati
Movable Type