Effect on loading a DC motor - load torque
The basic principle is simple - torque is proportional to armature current * magnetic flux. In a permanent magnet or shunt wound motor you can assume that flux is constant, so torque is just proportional to current. 

However this does not take into account internal friction, windage, and magnetic losses. When the motor is running free these losses cause it to draw a no-load current (Io). Subtracting Io from total current draw leaves you with the portion that produces output torque.

Power = rotational speed x torque. As more load is applied the motor draws more current, which increases torque. However as current flows through the windings their resistance causes the effective voltage to drop, so speed decreases. Below 50% rpm the power output will also decrease, reaching zero at stall.         

In a series wound motor the situation is different, because flux is not constant. With no load a series wound motor will speed up until friction and windage losses match the internal torque supplied by Io. This rpm could be very high, perhaps even high enough to destroy the motor. When a load is applied the resulting torque is proportional to current squared, because both armature and field currents contribute to magnetic force.