Browserquest is a game put together by Mozilla (the company behind the popular Firefox browser) and intended to show off WebSockets, a technology that allows constant communication between your browser and a server online. But outside of the tech, it's a nice little streamlined MMORPG. There's no story to speak of, but you can explore a beautiful world, meet NPCs, and kill creatures while upgrading your items.
Browserquest is also impressive because it's as cross-platform as these things get -- even just resizing your browser screen will change the way the HTML 5 game is displayed, so it works great on your big-screen monitor, mobile devices, or anything else you run it on. If you happen to be a coder, the full source is also available on Github, so you can see how it's all done, and even use that code to build your own games.
The graphics and gameplay are charming enough that you'll wish it was filled out more, but Browserquest already works as a template for what browser-based games might be like in the future.