While I have few complaints about the standard Android browser, these browsers have a little something extra that makes them stand out from the crowd.
Skyfire Web Browser 4.0
One of the best browsers for video and social sharing.
Dolphin Browser Mini
This is a solid, clean, highly customizable (upon set-up it lets you choose the function of your volume button, for example), tabbed, and fast browser. It has a very interesting ‘gesture’ feature that I find incredibly cool, if not massively practical.
Another browser with a nice, fast, tabbed user experience–less customizable than Dolphin, but easy to use and intuitive.
Firefox Browser fast & private
Very buggy on a Motorola Droid at this point, but has many interesting add-ons, the ability to restore all tabs if the browser crashes, and the option to sync tabs between your phone and other computers running Firefox.