The flight controller (or FC or sometimes just “board”) is the brain of the quad. It uses data from a gyro & an accelerometer (just like the ones in your smartphone) & input from the radio control link - It knows what the quad is doing & what it should be doing. The firmware running on the FC (usually either Betaflight, Raceflight or Kiss) keeps the quad stable by controlling the motors speed.
The F-number of a board denotes its power. F7 boards are the fastest & just starting to arrive. F4 boards are the norm. F3 boards are still a sound option but won’t run some of the latest features. F1 chips are now obsolete - steer clear.
Other features include All-In-One boards (AIO) that handle the power distribution for your quad, On-screen Displays (OSD) that display useful info in your FPV feed & SD card slots that record logs that help with tuning.