What is meant by damped and driven harmonic motion?

Damped harmonic motion

In the absence of any resistance forces (like friction and air resistance), most simple harmonic motions would go on unchanged forever. In reality, this doesn't happen, because there are resistance forces.
Damped harmonic motion - harmonic motion in which energy is steadily removed from the system.

There are three kinds of damping:

1.    Critically damped - the damping is the minimum necessary to return the system to equilibrium without over-shooting.
2.    Underdamped - less than critical, the system oscillates with the amplitude steadily decreasing.
3.    Overdamped - More than critical, the system returns slowly towards equilibrium.

Driven harmonic motion

Also known as forced harmonic motion, this is harmonic motion in which the system is given a periodic push. A perfect example is a person on a swing.

How the system behaves depends on how the frequency of the driving force compares to the natural frequency of oscillation of the system.
The most efficient way to transfer energy from the driver to the system is to match the frequency of the driving force to the natural frequency of the system, such as you do when pushing someone on a swing. This is known as resonance. At resonance, relatively small driving forces can build up to large-amplitude oscillations, just because energy is continually being injected into the system at just the right frequency

Damping is just like friction in the kinemetics as any body continues its state of motion untill unless an external force acts on it and definitely the force has to be in the opposite direction to the direction of motion.but if a body is moving on the rough surface then opposition of motion is done by the frictional force and this frictional force eats up the energy and rthe body stops simillarly if damping force is there it also eats up the energy of the oscillating body consequently the amplitude decreases and keep on decreasing and finally the body stops performing SHM