If you run the motor on 6v then its no-load speed will be about half what it would be at 12v, and its stall torque will be about half what it would be at 12v. [EDIT: I am assuming a permanent-magnet motor.]
Bear in mind that if you plan to control the motor from an Arduino, the control device you use between the battery and the motor may have a significant voltage drop, so the motor will get even less than 6v. The voltage drop is negligible using a mechanical relay, generally very small using a mosfet or mosfet-based h-bridge, and substantial using a darlington or an h-bridge based on darlingtons.