• Momentum is conserved in all collisions.
• Mechanical energy is sometimes conserved in collisions.

The mechanical energy (basically kinetic) is often lost to other forms of energy such as heat, sound, deformation, etc.  (Total energy is always conserved, it's just that mechanical energy may not be conserved.)  In other words, we have non-conservative forces.

If mechanical energy is conserved then we call the collision elastic.  Here we will see the objects bounce apart such that their relative speeds are the same before and after the collision.

If the two objects stick together, then we call this a perfectly inelastic collision.

Inelastic collisions are ones where some energy is lost, but the two objects don't stick together.  Elastic and perfectly inelastic collisions are useful models, but most collisions are somewhere in between these two extremes.

We can measure their "elasticity" by a ratio of the objects' relative speeds before and after the collision:

This is known as the coefficient of restitution.  For elastic collisions e=1 and for perfectly inelastic collisions e=0.
 
 

Keep in mind that these names don't represent any new physics, they're simply ways of classifying types of collisions.