What is the difference between an elastic and inelastic collision?

The most important rule for any collision in a closed system is that total momentum is always conserved. This means the total momentum before the crash equals the total momentum after the crash, whether the collision is elastic or inelastic. The difference lies entirely in the kinetic energy. In an elastic collision, the total kinetic energy before the collision equals the total kinetic energy after. In an inelastic collision, the final kinetic energy is less than the initial kinetic energy.

In the macroscopic, everyday world, perfectly elastic collisions are incredibly rare. The closest everyday examples are billiard balls clicking together or steel pendulums (like Newton's cradle) bouncing off one another. In these cases, the objects do not permanently deform, generate very little heat, and don't stick together. In physics problems, you can usually assume a collision is perfectly elastic only if the problem explicitly states it or if you are dealing with subatomic particles like electrons.

Almost all real-world collisions are inelastic to some degree. A car crash, a baseball being hit by a bat, and a dropped textbook hitting the floor are all inelastic because energy goes into bending metal, making noise, or creating vibrations. A special case is the 'perfectly inelastic' collision. You can easily spot this in a homework problem because the two objects stick together after the impact and move as a single combined mass.

The most common mistake students make is assuming kinetic energy is conserved in every collision. If you try to set initial kinetic energy equal to final kinetic energy for a car crash problem, you will get the wrong answer. Always start collision problems by using the conservation of momentum ($p_i = p_f$). Only use the conservation of kinetic energy ($K_i = K_f$) if the problem specifically tells you the collision is elastic.