1. Blown fuse. The first thing you should check is the fuse that powers the cigarette lighter and power plug. The fuse is located in the fuse box, which is usually under the dashboard on the driver's side. The fuse for the cigarette lighter and power plug is usually a 15-amp fuse. If the fuse is blown, replace it with a new 15-amp fuse.
2. Loose connection. The next thing you should check is the connection between the cigarette lighter and power plug and the wiring harness. Make sure that the connection is tight and that there are no loose wires.
3. Faulty cigarette lighter or power plug. If the fuse is not blown and the connection is tight, then the problem may be with the cigarette lighter or power plug itself. You can test the cigarette lighter by plugging in a known-good device, such as a cell phone charger. If the device does not charge, then the cigarette lighter is faulty. You can test the power plug by plugging in a known-good device, such as a lamp. If the lamp does not light up, then the power plug is faulty.
If you are not comfortable working with electrical wiring, you can take your car to a mechanic to have the problem diagnosed and repaired.