The rotor turns.
It has the magnets attached to it with a shaft running through. The construction of the rotor varies from one motor to another, but they all are basically put together in this manner. Even the outrunner type motors (Nippy, AXI, etc) are like this. Difference is the magnets are on a bell shaped part placing the magnet segments outside of the windings, vs. inner runners (Hacker, Aveox, etc) They have the windings placed inside the case surounding the rotor. Wish I had some pictures to post, it would make it simple to understand.

Thanks for the explanation on brushless current Matt, rogwabbit

Bill