HTML Validator is a Mozilla extension that adds HTML validation inside Firefox and Mozilla. The number of errors of a HTML page is seen on the form of an icon in the status bar when browsing. The details of the errors are seen when looking the HTML source of the page.
It's many times more useful with Dragonfly 9.2.1, 'cos the minimum error/warning count is down to almost 0 now. Before it was hard to "see the wood for the trees." Unfortunately Admin mode is still a bit hard to use due to a number of "built-in" errors, but time will address those I'm sure.
One thing to watch though, HTML Validator will often return a smaller number of errors/warnings than if you tested directly against W3C. Reason is that HTML Validator will give you just the first instance of a much repeated error, while W3C Validator gives you every one.
Both use same test, to the validity is still there. In fact for debugging XHTML, HTML Validator is preferable IMHO, due to the slimmer output, just so as long as you remember to do a search for all similar errors while you are correcting the listed ones.