Oscillations that occur when a body capable of oscillating is subject to an applied force which varies with time. If the applied force is itself oscillatory, a differential equation such as mẍ+kx=F0 sin(Ωt+ε) may be obtained. In the solution of this equation, the particular integral arises from the applied force. For a particular value of Ω, namely resonance will occur.
If the oscillations are damped as well as forced, the complementary function part of the general solution of the differential equation tends to zero as t tends to infinity, and the particular integral arising from the applied force describes the eventual motion.
http://www.acoustics.salford.ac.uk/feschools/waves/shm3.htm Animations and videos of forced oscillation and resonance.