I'll add to OOfffs (very well stated) post. An Aeropress uses a bit of pressure (you pushing it down) in the brew process to force the water through the grounds vs a French press where you're really just steeping grounds in hot water and then straining the brewed coffee when you push the grounds to the bottom. So in a way, the aeropress process is not dissimilar to how an espresso machine brews which is why it stands to use a finer ground. I even find you get a pleasant bit of creaminess/smoothness (like an espresso) with an aeropress that you don't get with a drip, pour over or French press.