Both the Aeropress and the Plunger (French Press or Cafeteria) are immersion brew methods, the Aeropress however, also uses pressure.
The Aeropress was invented by Alan Adler and was sold, until recently, under the same brand as Adler’s equally famous invention, the Aerobie flying ring. The tinkering inventor was trying to work out how to brew a single-serve of coffee in his large drip coffee machine and came to the conclusion that he needed to be able to apply pressure to the brewing, inside of an airtight chamber.
The most common way of brewing an Aeropress is the inversion method. The Aeropress is placed upside down, coffee grounds are placed into it, hot water is poured over them, the coffee is left to steep, the filter paper is screwed onto the top, and then the whole device is flipped over and the water is pushed through the coffee and filter into a cup producing a rich and sweet brew of coffee.
The Plunger is an immersion brewing method where hot water is poured onto the coffee, the coffee is left to steep, and then the filter disk is pushed down separating the grounds from the brewed coffee. The coffee should then be decanted immediately as the coffee grinds will continue to brew through the metal gauze filter.
If you are looking for the next level plunger, the inventors of the Espropress have created a better press through the use of a sealed chamber that separates the grounds from the coffee once the plunger is pushed down. In an Espropress the coffee can sit for up to an hour in the insulated plunger.
When you are making coffee for a couple of people, the plunger is perfect. When brewing coffee for yourself, the biggest benefit of the Aeropress is how easy it is to clean up after. As you unscrew the cap you can eject the coffee and filter straight into the compost, and with a rinse, the Aeropress is clean.
Further Reading: Is coffee a long-life product?