Total Validator appears to be the most up to date way of testing your Web site for accessibility. Of the current validation extensions, TV is the first to validate against WCAG 2 and has the most options to validate against several standards in one hit.
Download TV from addons.mozilla.org/en-US/firefox/addon/total-validator/.
You can select various doctypes for your validation, giving you a chance to validate against a new standard before changing the doctype in a page.
You can validate against WCAG at any level in the one test, giving you the chance to clean up your side for the measurable and achievable WCAG A before tackling the more complicated WCAG AA or AAA. Note that level AA and AAA are more complicated that level A mainly because there are a lot of subjective tests that cannot be performed by a computer. If you want a level AA or AAA check list, validate with Cynthia Says.
You can choose to use a local validation or a Web based validation. The free local version does one page at a time and is great for adding new pages. The Web based validation lets you validate 15 times per day, enough for adding a page or two each day, and includes screenshots for a variety of browsers. For a modest fee, US$40, you can buy the Pro version and validate whole sites as often as you like.
There are some other sites that have a larger range of screenshots from Web browsers, particularly mobile devices, but do not validate for accessibility.
TV is the first choice for accessibility validation and the web version gives you a basic set of screen shots.