What does it mean to "validate" a page? Why would you want to?

Web pages are layed out with text. To validate means to check this text against the agreed-upon standard. Usually it's done by entering the URL into the W3C's validator. You can see how this page did by clicking here.

It's important for a page to validate because it indicates that most browsers will display it properly. Many browsers display the same markup different ways, so a page may look very different from one browser to the next. Although no one can know all the possible ways a document could be interpreted, most users will have one of maybe a half-dozen common configurations, which is a much more managable pool to test against.