There are many ways to brew coffee and different brew methods can change the flavor of the coffee but how exactly?
Coffee body is one of the key elements to how much you enjoy the coffee and the primary way you can control body is through the filter.
But what is "body" in a coffee? Body refers to the mouth-feel of the coffee. A coffee with more body has more soluble lipids in the cup. These lipids are responsible for the aroma and flavor of the coffee and since coffee is intrinsicly bitter, more lipids equal more bitterness. Coffee filters can trap these lipids and prevent them from entering the final, brewed coffee. The more lipids you trap, the cleaner, "juicier," and sweeter the final coffee will be.
A French press has a very coarse, metal filter that primarly traps the ground coffee and allows a majority of the lipds to pass into the cup of coffee. This produces a very heavy-bodied coffee, often with a very distinct oil slick floating on top. That oil slick is caused by the oily lipds floating on the water. Brewing Tip: always decant your coffee from the press around four minutes after adding the hot water to the grounds. Otherwise your coffee continues to steep and will become overextracted, bitter, and unenjoyable.
On the opposite end of the filter spectrum is the Chemex paper filter, which is a very dense, thick paper filter. The Chemex filter traps a lot of the lipids and the resultant cup of coffee has a very light body that is clean and juicy. Some say (and I would agree) that the Chemex is a great way to bring out the unique varietal flavors of a coffee by reducing the bitterness of the lipids that can sometimes distract from the more subtle and sweet flavors present in a coffee.
Experiment with different brew methods and filters! You'll find that different coffees are better when brewed in specific ways and it all boils down to what you like in a cup of coffee.