It's a semi-anonymous currency (not quite as anonymous as cash, but much better than PayPal or whatever). New currency is created and added to the system by people solving some sort of cryptography something something whatever, which high end video cards happen to be good at apparently. It's complex.

It's not widely accepted, but there's a list of places that take Bitcoin somewhere. It's also apparently quite popular in the deepweb, Silk Road and such, since it's relatively hard to trace transactions or link them to actual people.