Outlets are hot but nothing turns on
The non-contact tester isn't telling you the whole story

A circuit requires 2 functioning wires. One of them is "hot" which you are familiar with. However power has to return to complete the loop.  This is very often done with "neutral", and neutral is so near ground that it won't register with a non-contact tester.  Normally. 

However, if hot is working and the socket is not, there's a good chance neutral is broken.  You can use the non-contact tester to find it. 

You need a load you can plug in that is definitely turned on.  Something very simple like a heater or an incandescent light.  You plug that into the problem socket.  Now, anywhere past the neutral break, the neutral will test out as "hot".  When you find the first outlet where this is true, check it.  Also check the last "good" outlet before this point.