When you apply the power, a current flows in the armature I = V/R where R is the armature resistance. The field produces a torque proportional to the current and the motor starts to accelerate. Once the armature moves, the motion and field induce an EMF in the armature which opposes the supply voltage and hence the current, so the current is now I = (V-E)/R. The power dissipated as heat, P = RI^2, and the power used in accelerating the armature, P = EI.

As the back EMF is proportional to the speed of rotation, the current will diminish as the speed rises, and the acceleration diminishes with it. Eventually you will reach a point at which E is almost equal to the supply voltage, and EI is equal to the frictional losses in the motor. The armature will then stop accelerating. If you now apply a load torque to the motor it will start to decelerate, and as it does so the EMF will reduce, the current will increase, and the torque delivered by the motor will increase. Eventually the torque delivered balances the torque from the load, and the armature will stabilise at a new speed lower than it ran with no load. The power delivered to the load will now be EI minus the frictional losses previously mentioned.

The above is true for permanent magnet and shunt wound motors, but series wound motors are slightly different, they will continue accelerating indefinitely in the absence of a load because as the current decreases, so does the field.