This is the system for a car with an automatic transmission. A black wire with a red stripe comes out of the fuse box, goes out through the firewall, then dives down to the transmission. On the side of the transmission, where the shift linkage is attached, is the neutral safety switch. This switch does two things. It prevents the car from starting when it's not in Park or Neutral, and, it sends power to the reverse lights when the shifter is in the Reverse position. The black wire with the red stripe goes into the neutral safety switch and comes back out of the switch, goes back through the firewall, and the goes to the back of the car with the tail light feed harness, which runs from under the dash, then underneath the driver's side door sill plate, and back into the trunk area.
A manual transmission car has that black wire with the red stripe coming out of the fuse box, and then running underneath the carpet and down through a hole in the floor right behind the shifter, where it goes into a reverse light switch which is positioned in such a way that, when the shifter is in the reverse position, the shifter arm pushes the button on the switch. The black wire with the red stripe then comes back up through the floor, back out from under the carpet, and then joins up with the tail light feed, and goes to the trunk area, just like on an automatic car.
The most common problems with this system are either that the bulbs are burned out, or the ground wire from the reverse light housings have been disconnected or cut, or just got all corroded and funky-looking, or, on an automatic car, the neutral safety switch has gone stupid on you. On a manual tranny car, the same things could happen with the bulbs or the ground, it's just the reverse light switch that goes stupid instead of the neatral safety switch, since it doesn't have a neutral safety switch. If the lights stay on all of the time when the key is in the 'On' position, that is almost certainly a defective switch.