Light fixture in basement with a light switch that has an outlet next to it ........light won't work now
If you replaced a light fixture and when you turned it on it started to buzz then went out. Then let's go tho the obvious. You had a short in the wiring at the fixture. So first turn off the breaker to the fixture. Remove the fixture and look inside your box. I am guessing that you will find a wire burnt in half or at a splice or it may be wired incorrectly. If that fixture has a ballast or driver you can assume that the miss wiring has burnt it out and needs replacing.

You said you had a wire coming from the receptacle to the switch. You say the a conductor is coming from the receptacle and it didn't trip the breaker. I am assuming that would be your hot. Check the switch by using a continuity tester touch both posts of the switch and see if you can get it to ring when you turn the switch on.  

You also run a ground to the switch and connect the fixture to the ground to eliminate a possible shock hazard.  

Good luck and stay safe.