Conservation of momentum holds for a system when there are no external forces on the system. Since momentum is a vector, we need to check the net force in each direction, and if the net force is zero then the momentum for that direction is conserved.
The primary examples in intro physics are collisions (between blocks, disks, sports equipment, bullets and targets, etc) and also frictionless surface problems like ice skaters who push away from each other or people in boats that push away from each other, etc.
Conservation of energy in a system occurs whenever there is no external work done on the system. Typical causes of external work are friction, air resistance, or normal pushing / pulling.
Momentum is conserved in all collisions. Kinetic energy is only conserved in elastic collisions. In the case of inelastic collisions, kinetic energy is lost due to deformation of the objects involved in the collision.
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