Before one can troubleshoot trailer lights it is important to understand how they work. Trailer lights are designed to function in the exact same way as your vehicle's tail-lights. They are attached to the wiring from your vehicle by means of a connector device extended alongside the vehicle's trailer hitch. There are three wires; a common ground wire, and one hot wire for each trailer light. More advanced rigs may have more elaborate wiring setups, but these are the bare-bones basics that are dealt with when they're towing a U-Haul or a boat trailer. Start at the connector. Check to see if it's loose or corroded. If so, it needs to be cleaned and repaired. If everything looks good, then it's possible your ground wire has come loose. The metal of the trailer is actually used as the grounding point, so if the wire has lost connection with the trailer, the circuit is incomplete and none of the lights will work. Follow the length of the ground wire and make sure it terminates securely against the metal of the trailer.
If only some of the lights work, either you've got a few burnt light bulbs or one of your hot wires is faulty. You'll need a voltage meter to determine this. First check the socket with the meter to see if any electricity is getting through. If it is then detach the connector, scrub out the inside of the light socket for good measure, then fit a new light bulb into place. If no electricity is reaching the socket then follow the wire. Make sure the wire isn't broken anywhere. Pay particular attention to any patches where the protective plastic may have come off. If the metal of the wires touch the trailer it will cause a short circuit, preventing the electricity from reaching the socket. If the wires are whole and undamaged, but have no current running through them; test the sockets of the connectors. If the connector from the tow vehicle has current while the end from the trailer doesn't, the trailer's connector needs to be replaced. If the connector from the tow vehicle has no current, then the connector was not installed properly and a new one must be installed.
If neither the tow vehicle nor the trailer lights function, it's likely that a fuse has been blown. Check the fuse box in the driver's side foot-well and replace the fuses corresponding to the taillights. It's possible that the lights will function for a short time before stopping again. If this happens then a short circuit is causing the fuses to blow. This can happen at any point along the wiring, both in the tow vehicle and the trailer. However if the fuse does not blow when the trailer is disconnected it means the trailer will have to be rewired. To avoid further possibility of damage to your vehicle's electrical systems it's best to have the rewiring done by a professional.
