Outlet with power won't supply power to anything plugged in
1: You should really use a meter instead of those plug in devices - they do not tell you much. 

A cheap analog meter might cost $5 and make sure it is analog - a gauge with a physical needle. I know the clerk at a store will tell you the goods on the digital but skip that the analog meter will measure correctly with no phantom voltages, the digital unless it has Lo-Z will not measure accurately.

Is it just this one outlet or all the outlets on that circuit breaker circuit ?
If all outlets behave the same way - Replace the Circuit Breaker - it is defective. If only one outlet is defective read on.

Test Method 1:
Using that meter you can measure for 120VAC between the hot and neutral connections and also test hot to ground. Anything less than 110 VAC means you have a problem with that circuit.

Test Method 2:
Place an LED night light in the socket and see if it lights or even better a cell phone charger that has a little power light indicator to show that it is 'working' (don't plug the other end into your cell phone!) , if the charger indicator light is on or the LED night light is on - it does mean you have power there but does not tell you how much. Since you never said if one or all outlets on this circuit breaker circuit are affected, this test merely declares the possibility of a misbehaving breaker - Led charging light lit (lo current draw - probably a bad breaker). 

If Test Method 1 Showed less than 110VAC or Test Method 2 the light on a cell phone charger lit up or even did not light - you have one of Five possible things going on.


A Bad Outlet. 
A Bad Connection at the outlet.
A Bad Connection up stream from this outlet. 
A Bad Connection in the Circuit Breaker Panel. 
A Bad Circuit Breaker.  


For items 4 and 5 I would check the other outlets controlled by that circuit breaker to see if they behave the same way. If they do 4 or 5 is your problem. If not 1 through 3 is your answer.