Question

What is the physical significance of momentum. Like velocity gives us how position is changing with respect to time, what does momentum of an object show?

Answer 1

The faster an object is moving—whether it be a baseball, an automobile, or a particle of matter—the harder it is to stop. This is a reflection of momentum,

Consider the term "momentum" outside the world of physics, as applied, for example, in the realm of politics. If a presidential candidate sees a gain in public-opinion polls, then wins a debate and embarks on a whirlwind speaking tour, the media comments that he has "gained momentum." As with momentum in the framework of physics, what these commentators mean is that the candidate will be hard to stop—or to carry the analogy further, that he is doing enough of the right things (thus gaining "mass"), and doing them quickly enough, thereby gaining velocity.

Try to match this answer with some games like billiards/pool :)
If you want to describe a combined effect of velocity with which you struck the ball and the mass of the ball you use the word momentum!
Hope you get it :)

Answer 2

It is significant (moreso than its components of mass and velocity) because it is always conserved, whereas mass and velocity is not.
It's also a thing that pops up a lot in quantum mechanics (again, as opposed to its components of mass and velocity ( http://en.wikipedia.org/wiki/Uncertainty_principle)).

Answer 3

* http://en.wikipedia.org/wiki/Uncertainty_principle