The browser-based compiler
can now run code, thanks to Martin and his ethereumjs-vm package!
Everything is in memory only and the interface is still really crude. For example it is not possible to give any function arguments yet, so all functions that take arguments will fail.
To start, click the "create" button, which will deploy the contract. Once it is deployed, you can use the buttons that appeared to call individual functions on the contract.
If you change the code (causing it to be recompiled), these buttons will vanish, so be careful.