This is a very simple list for simple sites.

More serious sites will have a much longer list.

Test all supported browsers.

Test with devices: iPhone, iPad, serveral Android phones (Androd 2.3, 4.1...), Windows Phone

Test with several screen widths (responsive design)

Test with ad blocker / Ghostery

Test for SEO

Test title of every page

Test internationalization

Test accessiblity (good contrast, no small print etc.)

Test with reader software

Test alt attributes

Test low bandwidth

Test with keyboard only

Test with high load (many requests per sec)

Test caching headers

Test minification and merging CSS / JavaScript files

Test compression of textual files

Test favicon and touch icon

Test charset

Test lang attributes

Test HTML5 form features on devices

Test offline (when connection gets lots in train for example)

Test image and description when the site is added in a Facebook message

Etc. etc.