Physicists love to look at something complicated and find in it, something simple and familiar. Here is an example. If you flip a baseball bat into the air, its motion as it turns, is clearly more complicated than that of a non-spinning tossed ball, which moves like a particle. Every part of the bat moves in a different way from every other part. It is a system of particles. But if we look closely, we find that one point of the bat moves in a simple parabolic path. That is nothing but the centre of mass.
What is the importance of knowing the centre of mass of an object? Once we know the centre of mass of a body, one can balance the body by placing a support exactly beneath the point.