Kindle "keyboard" devices running firmware 3.x (including jailbroken Kindle 2 upgraded to firmware 3.x) support the ES3 version of JavaScript and no HTML5 layout features, similar to what was supported by Internet Explorer 8.
Newer e-ink Kindle devices, like the PaperWhite, Oasis and Voyage running firmware 5.x support the ES5 version of JavaScript and some HTML5 layout features, similar to an original iPad running iOS 5.0.
You can inspect the source code for the browser, which Amazon supplies in their open source compliance download package. It is based on the WebKit sources circa 2009 with some customizations for e-ink rendering. The browser passes the Acid 3 standards compliance test with 100 out of 100 tests passing and almost totally matching the reference rendering.
The main issue with the browser on older Kindle devices is the outdated SSL certificates means any site that requires SSL (https instead of just http) will not load. Amazon did issue an emergency fix, which all users should apply so they can continue to both browse the web and connect to Amazon's digital storefront.