Conservation of momentum is a fundamental law of physics, derived from the famous three laws of Isaac Newton. It states simply that in a closed system of objects, the cumulative momentum within the system remains constant. The general formula to explain the principle is
m1v1 + m2v2 = m1v3 + m2v4
Where m1 and m2 are the masses of two objects colliding, v1 and v2 are the initial vector velocities, and v3 and v4 are the vector velocities after the collision.
So first the logical derivation. Newton's first law states that a body in motion will remain in motion (and a body at rest will remain at rest) if it is not acted upon by other forces - that is, its momentum (mass times velocity) will remain constant. So in a system with only one object, we can see that the conservation of momentum is true. In a system with two objects moving parallel to each other (or indeed, any number of objects in parallel) momentum will also remain constant. Of course, there are very few systems in which all objects move in parallel (and the ones that are tend to be very dull to watch.)
Newton's third law is what makes thing interesting. It states that for every action there is an equal and opposite reaction. This principle ties directly to the effects of collisions in a closed system, and the effects of those collisions on the momentum of the objects that are colliding. In short, when two or more objects collide, the sum vector velocity of the objects will remain the same. So if the momentum of one object is lowered. the momentum of the other objects has ipso facto increased.
The classic example in physics lectures of the conservation of momentum are pool balls on a pool table. Let's imagine a pool ball striking head on (that is, at a 0 degree angle of incident) another pool ball which is at rest. (Sidenote: This type of collision is called an elastic collision, because the balls don't absorb any of the kinetic energy. Imagine if the two balls were made of Jell-o instead. That's an inelastic collision, because it wastes energy through the object's deformities.) First, we all know what happens - the ball that strikes comes to a halt, and the ball at rest goes out in whatever direction it was struck towards. The reason? Conservation of momentum. Per our formula above, m1 is in theory equal to m2. v1 is however fast the first pool ball was going when it struck the resting ball, and v2 is obviously 0. By transferring its kinetic energy to a mass of equal weight, and through conservation of momentum, the first pool ball's velocity (v3) is reduced to 0, while the formerly resting ball's velocity (v4) rises up to equal v1 - thus keeping the formula even on both sides.
This is of course the simplest variation of the conservation of momentum. In a scenario where an object strikes an object half its mass (m1 = 2 * m2), the velocity of the struck object will be twice the velocity of the original striking object. Similarly, if an object strikes another at an angle other than head on, (the angle being the angle between the two objects' centers of mass), then we must consider the final vector velocities and not merely our common shorthand for velocity, speed. It's also important to note that conservation of momentum holds true even when objects break apart (a bullet hitting a watermelon) or when objects fuse together upon contact (a hydrogen atom combining with another hydrogen atom.) Physicists are often required to calculate the momentums of multiple objects (such as atoms) acting within 3-dimensional systems (such as a beaker) in order to gather evidence of a reaction or effect. Knowledge that the momentum of the system is conserved allows them to do just that.
You can observe the conservation of momentum yourself in Newton's cradle - that's that little set of suspended orbs that you can swing into each other. Depending on how many you select to swing, the effect observed is that the same number of orbs will be "kicked up" off the end, and when they themselves return to earth, the original orbs will be kicked up once more. Repeat ad finitum (friction for the win!).