Core aeration is the removing of plugs of dirt from the turf. This allows air into the ground, which promotes healthy soil and prevents soil compaction.
Core aeration is different spike aeration, which actually creates compaction in the soil while allowing air into it. The process requires a heavy machine that is able to poke a hole 3-4 inches into the ground. It pulls out the core of dirt, and drops it on the turf. These plugs will move back into the soil with regular watering/rain, or by running a mower over the plugs on a mulching setting. The whole yard is gone over with the aerator, pulling plugs and allowing air into the soil.
Core aeration promotes a healthy root system in the same way that pruning a rose bush promotes healthy, beautiful blooms. When the aerator tines go into the soil, it also cuts some of the turf roots. This has a pruning effect, allowing new growth in that root, which in turn means healthy, beautiful turf.