Damping refers to the phenomenon where an object undergoing oscillations experiences resistive forces, such that it causes the amplitude of oscillations to decrease or decay. The three types of damping are then defined by the way the amplitude decay with time.
Light damping, from the name, refers to the condition where weak resistive forces act on an object exhibiting simple harmonic motion, such that its amplitude decays exponentially over a relatively long period of time. Because of that, the object is able to undergo multiple oscillations before it stops.
Critical damping refers to the condition where an oscillating object is forced to return to its equilibrium position as quickly as possible. This happens when the resistive force(s) is extreme and sharp, enough that the object does not move back and forth, and instead halts to a stop in a very short period of time.
Heavy damping, once again from the name, refers to the condition when an oscillating object experiences large resistive forces, such that it slowly returns to its original position without covering multiple oscillations. In that regard, heavy damping is different from critical damping because here, the object halts to a stop over a longer period of time.