What determines how much power the motor 'draws'?
Basically, the load on the motor determines the current.  There are two main things to keep in mind: 

1) The motor when turning generates a speed-proportional voltage, referred to as the back-emf, that opposes the applied battery voltage: E = Ke * speed, where Ke is the back-emf constant.

2) The torque produced by the motor is proportional to the current: T = Kt * I, where Kt is the torque constant.  (Note: if SI units are used throughout, Ke and Kt are identical.)

The motor windings have resistance, so the current flowing causes a voltage drop.  The motor will run at a speed where the back-emf plus the voltage drop due to the current in the motor windings equals the battery voltage, and where the motor torque equals the applied load.  

Without any external load, the motor torque only has to counter rotational losses, such as bearing friction.  So the current is small, and the back-emf is close to the battery voltage.

If a load is applied to the motor shaft, the motor will slow a little and the back-emf will decrease proportionally.  This increased discrepancy between the battery volts and the back-emf forces more current in the windings, consequently the shaft torque increases.   The speed drops and the current increases until the motor reaches equilibrium at a new operating point.  

A plot of speed vs torque would give a sloping line; the slope of the line is indicative of the winding resistance