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In an elastic collision, in addition to the total momentum being conserved, the two bodies do no work on one another during the collison, so that the total kinetic energy does not change, that is, Ki = Kf
In a completely inelastic collision, the bodies merge or stick together upon collision. While the total momentum is of course still conserved, the bodies do work on one another, so there is a big change between initial and final kinetic energy.
Under ordinary conditions in nature, collisions between two atoms, or between two molecules, are elastic. The behavior of gases is an example of what appear to be hard elastic spheres colliding elastically with anything they encounter.
