Unplug the lamp or turn off the power to your home if you are checking a light socket that is connected to the internal wiring of your home. Remove the light bulb. Take off the insulating sleeve on the light socket. Attach the continuity tester clip to one of the blades of the power cord -- these are the prongs that plug into the wall.
Put the probe to the silver screw of the socket. Put the other probe on the brass screw. Attach the continuity tester wire to the other plug blade. Touch the probe on the silver screw.
Touch the tester wire to the brass screw. One of the two combinations you performed should have caused the continuity tester to light up. If neither combination works, the power cord is the problem and needs to be replaced if possible.
Attach the tester wire to the collar -- what the light bulb screws into -- and touch the brass screw with the probe. Flip the light switch on and off. If the continuity tester does not go on and off, there is a problem with the light socket itself. If it does go on and off, current is flowing through it properly.