What is the significance of momentum?
Momentum is what makes it "tough" to stop moving things.
If you stand with an apple in each hand, then you feel their weight. If you throw one of them upwards and catch it again, it feels "heavier" in that moment. What you feel in addition to the weight is the momentum.
Stopping a motion can be an easy or tough task depending on the momentum, which encompasses both the speed $v$ to decelerate from as well as the mass $m$ that resists this deceleration.
At the same time there happens to exist a conservation law regarding momentum. All momentum before equals all after any event; momentum is always conserved $\sum P_{before}=\sum p_{after}$. So apart from the physical significance of momentum, it also happens to be a very useful tool.
That's why we see it and learn about it and use it all the time.
Momentum is the conserved Noether Charge that corresponds to the invariance of the Newtonian dynamics Lagrangian under spatial translation. In less technically dense language, this means that momentum is a conserved quantity whose conservation arises because physical laws (or a particular, "Lagrangian" formulation of them) are invariant to a shift in the origin of the co-ordinate system that the laws are written in.
So as in the other answers, you can only change a body's motion state by having it transfer the change of momentum to you since the total system's momentum has to be conserved; you feel this transfer of momentum as a reaction force acting on you oppositely to the force you impart on the body. And every time you feel this force, you can think of the phenomenon ultimately as an expression of the principle that our physical laws don't care where we put the origin of our descriptive co-ordinate system.
For me, this is the deepest meaning for momentum, as I describe further in my answer to the Physics SE question What is momentum really?. I also discuss there some other slightly different usages of the word "momentum" in physics; the other usages are related to mechanical momentum in that they are defined analogously, but need not co-incide with it.
As the other answer says, the momentum is a "measurement" of the "inertia" that the body or the system of particles has.
My first teachers wanted us to call it "quantity of movement", because that's mostly what it is. (finally, the word mmentum conquered everything haha).
Think it this way, it's the velocity multiplied by the "mass", which is, after all, related to the number of particles of the body. It's the total amount of movement in that system. And then, as philosophers said, "the total amount of movement is constant in the Universe", it was finally true.