Newer versions of Chrome have an easier option for debugging with no need to install the Android SDK. I've written an updated guide here.
Fixing bugs in web applications or websites can be a pain in Android devices since there's no obvious way to access the usual webtools (Chrome dev tools or Firebug, for instance).
However, if you're not using Chrome on your phone and are stuck with the stock browser (or have a bug occurring only in the stock browser) then remote debugging won't work since it only works with Chrome on both devices.
To view the console messages, you'll need to have the Android SDK set up; the easiest way to view the console is through Eclipse (or similar) in the DDMS view. In Eclipse, this is under
Window > Open Perspective > DDMS
Some more links: