Electricity must make a complete circle, from a point of origin (where it enters the house from a utility supplier) and back again. Modern household electricity uses a three-wire system, a "hot" wire with black insulation that carries the current, a white "neutral" wire that returns it and a bare copper wire that shifts current to the ground if the circuit is interrupted.
Several things can cause an outlet to quit working.
Check for GFCI
The first thing to check if an outlet quits working is the type of outlet. A ground fault circuit Interrupter outlet can cease to function if the circuit feeding it is disrupted in any way, even temporarily.
A GFCI outlet has a miniature circuit breaker inside that will stop the flow of current if there is any problem. A button on the face of the outlet will pop out.
If the button is out, push it back in and see whether the problem is cleared.
Inspect the Breakers
The second step is to check the breaker panel where power comes into the house. A GFCI break will sometimes trip the breaker controlling that whole circuit, so it must be reset.
If all breakers are good (no red showing beside them), test other outlets in the house. Circuits are not necessarily divided by room; two outlets in a room could be on different circuits, and the one not working could be on a circuit connected to outlets in other areas.
Remove the Receptacle
If tests confirm only that one outlet is not working, remove that receptacle to check it. Turn off the basic power, with an individual breaker if you can identify the correct one or with the master breaker to block all power.
Remove the faceplate, which will have one or two screws, remove the top and bottom retaining screws, and then pull the outlet straight out of the box in the wall. If there is no box, call an electrician.
Inspect the wire connections to be certain they are secure. Some "stab-in" connectors are prone to failure; screw connectors can sometimes work loose so that wires don't make good contact.
Test the Wiring -- Carefully
You can test the wiring with a circuit tester, but don't try this unless you have some experience. It can be a dangerous operation; proceed with caution.
Turn the power back on, and touch one prong of a circuit tester to each circuit wire on the outlet. Hold the prongs by the insulated wires connecting them to a light.
If the light lights, current is flowing and the outlet itself is bad and must be replaced.
Replace the Outlet
Replace a nonworking outlet that has current going to it by removing the black "hot" and white "neutral" wires from their terminals and the bare copper wire from a green screw. Use the existing loops on the wires, if possible; otherwise, strip a half inch of insulation from the end of each wire, bend it in a hook and secure it under the screws, black to brass, white to silver and bare to green.
Of course, be certain there is no current to the site before replacing the outlet.
Call an Electrician
If there is no current flowing to the outlet but other outlets and switches are working, call an electrician. There is a problem somewhere that needs professional attention.
Also call an electrician if there are only two wires feeding the outlet or if the insulation on those wires is frayed or decayed in any way; modern wiring uses plastic insulation, but some older wiring may have a fabric covering, which can decay over time.