Why Is My Electrical Outlet Not Working But My Overhead Light Is?

I'm experiencing a similar problem. I turned on my electric kettle and heard a 'click' sound which I presumed was the breaker being tripped as the kettle didnt work. I checked the breaker box and nothing was tripped. I tested the kettle in other outlets and it works fine, and have deduced that the outlet itself is not working at all.
I changed the outlet and re-wired it correctly only to find that it still doesnt work. Tested the outlet with a circuit tester and nothing lights up; however up until recently everything worked fine.
Is this a simple case of bad wiring finally giving up, or is there something more sinister which will require more costly attention? Please advise, thanks.

They may not be on the same circuit breaker.
Have you checked your breaker box to see if any of the switches are flipped?
You may have something plugged in that has bad wiring or overloaded the circuit.
Sometimes the circuit breaker itself will wear out and need to be replaced.