Browserscope (http://www.browserscope.org/)This site allows you to test the capabilities of your browser as well as see the capabilities for other browsers people have tested. This does test browser capability, but also gets a little more down into the details than other sites on this list. For example, it you are wondering how many concurrent network connections a particular browser supports, then this is the site to go to.
HTML5 Test (https://html5test.com/)More and more browsers support HTML 5 these days, but there are still important differences out there. This site will not only test your browser, but compare different browsers to each other so you know what capabilities are and are not supported.
Can I Use (http://caniuse.com/)Another site about what features are available in what browsers. In this site, you select the feature from the main page and it then takes you to a page showing you what browser versions support that feature.
Modern IE (https://www.modern.ie/en-us)This site by Microsoft allows you to scan a page to see if you are using any old or deprecated features that you should be uplifting. What is even better is they have an interface to Browserstack which will take screenshots of your website right from a webpage.
Google PageSpeed Insights (https://developers.google.com/speed/pagespeed/insights/)Web performance is a big deal, and PageSpeed Insights quickly scans a target website for front end best practices. You will get a score for both mobile and desktop page speed as well as mobile usability. In addition, you will get specific recommendations about how the performance of your site can be improved, along with links to Google's documentation explaining exactly what needs to be done.
The online version of PageSpeed Insights can only target publicly available web sites. You can also check out the Chrome Extension. One of the nice things about the extension is that for things like optimized images, the extension will provide you with an already optimized version.