I've noticed that a lot of pages use "alt" tags in their images, and I'm not really sure what the purpose is. Do I need to use them?

Alt tags are used to provide alternate text for your images.

There are several reasons for including alt tags in your pages.

You can use the alt tags to provide more keywords, which helps with website rankings in search results.* 

If your picture is slow loading, the alternate text lets the user know what the picture is, while waiting for it to load

Similarly, if your image is hosted on a different server, and that server goes down, the image will not load at all, so the alternate text will be very important

Perhaps most importantly, alt text provides valuable information for vision impaired visitors, who cannot see the images, and so rely on text.

* 2016 update: this is not a terribly helpful technique any more. Because alt tags were generally not visible to the public, webmasters started using them as a way of spamming keywords into the search engines, which resulted in the search engines more and more ignoring these tags.

It's still good to provide alt tags - just not for this reason!