in an inelastic collision kinetic energy (KE) is not conserved because some of the original KE goes into heat, deformation, etc. For example if you have a putty ball and it rolls in the +x direction into a mass on a frictionless surface (a one-dimensional completely inelastic collision) we can imagine that the putty ball squishes into a disk. So the KE is not conserved.
The momentum of the ball decreases, just as its kinetic energy does. Conservation of Momentum applies to the system of the ball plus the mass that it collides with. The mass gains an amount of momentum that is exactly equal to the amount lost by the ball.
This follows directly from Newton's Third Law of Motion. During the collision, the force the ball exerts on the mass is equal in magnitude and opposite in direction to the force the mass exerts on the ball. The two forces act for the same time period. Change in momentum = impulse = force x time. Therefore the change in momentum is also equal in magnitude and opposite in direction for the ball and the mass.