How can I power an NXT motor without the NXT brick?
I think the simplest way is to use an NXT Converter Cable (Available from Bricklink). 
This has an NXT connector on one end and the standard 9v connector on the other. You can connect the 9v connector to any 9v battery box, (e.g. Battery Box with Switch (9V)) and, depending on the polarity, power the NXT motor forward or backward.

Now, the issues:


The speed of the motor is determined by the strength of your
batteries. With new batteries the motor will run at full speed and
you have no ability to vary this speed. Off/forward/backward are your
only options.
There is no way to receive feedback from the motor, so no way to
control the number of rotations as you can when connected to an NXT.