A biofilter is a component within an aquaponic system that catalyzes the conversion of ammonia to nitrates and then to nitrates by providing an ideal environment for beneficial bacteria. There is no "mechanical filtration" as the name suggests.
Water from the fish tanks enter into this biofilter with Ammonia from the fish. The component then bubbles this water vigorously with some form of media that creates surface area for bacteria. Because this bacteria thrives in aerobic environments, the conversion of ammonia to nitrates happens significantly quicker. These nitrates then get used up by the plants to grow your delicious food.