FPV stands for First Person View, and is a type of drone flying that uses a camera on the drone to send a live video feed to a pair of goggles or a monitor. This allows the pilot to see what the drone sees, and fly it as if they were in the cockpit.
Some common reasons to fly FPV:
Freestyle
Flying for fun and doing tricks.
Racing
Flying through gates and obstacles. Like Formula1 in 3D.
Cinematic
Flying for scenic views or one-shot indoor videos. Mostly used for professional work, like real estate.