inelastic collision
In an inelastic collision the objects hit and some of their motion energy disappears — into heat, sound, a dent, a crunch. Drop a ball of putty on the floor and it just goes splat, keeping none of its bounce; slam two cars together and the kinetic energy goes into twisted metal. This is what most real, everyday collisions are.
The precise statement: an inelastic collision conserves momentum but NOT kinetic energy. The total m times v before equals the total m times v after, just as in any collision, but the total 1/2 m v^2 afterward is smaller than before. The missing kinetic energy has not been destroyed — energy overall is still conserved — it has merely been converted into other forms such as thermal energy, sound, and permanent deformation.
Inelastic collisions sit on a scale of 'how much energy is lost', measured by the coefficient of restitution e (between 0 and 1). A superball is nearly elastic (e close to 1); a lump of clay is at the far end. The extreme case, where the objects stick together and lose the most kinetic energy possible, is the perfectly inelastic collision.
A 4 kg lump of clay moving at 5 m/s hits a 6 kg lump at rest and they stick. Momentum: 4 times 5 = (4+6) v, so v = 2 m/s. Kinetic energy before = 1/2 times 4 times 5^2 = 50 J; after = 1/2 times 10 times 2^2 = 20 J. A full 30 J — more than half — went into heat and squish. Momentum was still perfectly conserved.
Momentum in equals momentum out; the missing kinetic energy warmed the clay.
Losing kinetic energy does not break the law of conservation of energy. Energy is only transformed, not lost; 'inelastic' refers specifically to kinetic energy, the energy of visible motion, not to energy as a whole.