Aeration is when you loosen or puncture the soil in your garden to allow increased water to penetrate, as well as air permeability.
To aerate the soil, you can simply cultivate around new plantings with a trowel, or in a lawn, using a soil corer to remove small cylinders of soil from the dirt. For a more thorough aeration you can turn over a whole garden bed of soil with a shovel and break up the clods.