The back-up light switch controls your reverse lights. When you move the gear shifter into reverse, it physically closes this switch. This completes the circuit, sending power to turn on the white reverse bulbs at the back of the car. On manual transmissions, it is a simple threaded plunger switch screwed into the gearbox case. On automatic cars, it is built into the neutral safety switch on the side of the transmission housing.
If the switch fails, your reverse lights will not turn on when backing up. This is a safety hazard because other drivers will not know you are reversing, and it will fail safety inspections. In rare cases, the switch can short out, leaving the reverse lights stuck on constantly.