Momentum is a measure of how hard it is to stop a body from moving. This depends both on the how fast the body is going and how massive it is so we have momentum p = mv where the bold type indicates vectors.
Newton's second law in its most complete says Force = rate of change of momentum. So long as we work onlly with fixed mass systems this reduces to the simple form.
So a force is required to alter momentum and different forces cause different rates of change. We quantify this with Impulse. Impulse = change in momentum = force×time.
Since a force is required to alter momentum, any system on which no net external force acts has a constant momentum. The most common example of this is a system of colliding bodies. The bodies exert forces on each other but no net external force acts on them. In that case the total momentum is conserved, momentum before = momentum after.
Since that is a vector statement it means that each component of momentum is separately conserved.
Remember, momentum is conserved in any system in which no net external force acts. Examples include collisions on air tracks, cars colliding, a moon circling a planet, a rocket and its exhaust gasses.
Momentum is conserved in all collisions, unless one of the objects is tied down in some way so that an external force acts.
Mechanical energy is rarely conserved in collisions, except collisions among point particles. We call the rare collisions in which energy is conserved Elastic Collisions.
Any collision in which mechanical energy is not conserved we call an Inelastic Collision. The most extreme example of this is a collision in which the two colliding objects stick together. We call such a collision Totally Inelastic.
We can find the final state of a 2-body 1-D collision if we are given the initial state and told how elastic the collision is. This is measured by the coefficient of restitution e = (relative velocity after collision) / (relative velocity before).
We normally solve such 2-body 1-D collisions by solving the two equations
m1u1+m2u2=m1v1+m2v2 and v2-v1=e(u1-u2)
where u1 and u2 are the initial velocities and v1 and v2 the final velocities. For the special case of an eleastic collision e=1.