What causes the 60cyle hum when my peavey amp is on?

Turn your guitar away from the amp's speakers. The pickups on the guitar are feeding the amp's signal back into the amp creating a feedback loop, especially with the gain button on. Loud volumes cause this looping path between the amp speakers and guitar pickups. You may need to reduce amp volume also.

Most of the time amplifier hum is caused by the outlet that it's plugged into, maybe no ground or there's other appliances on the same circuit causing interference? And sometimes it's as simple as a little internal feedback, certain C notes on the Guitar cause distortion with tube amps, the best fix is a $100 power conditioner like made by Monster cables or others.... hope this helps?

Numerous possible issues with your description. I would start with the tubes, and check if not one or more has faulted.

Sounds like a bad power supply filter cap...the buzz you're hearing is the 60 cycles hum that filter caps reduce or (hopefully) eliminate altogether. It may also be related to the tubes but finding the bad one will require some work. Any friends have this same amp? If so, try swapping one tube at a time and see if you get lucky. If this amp is new take it back ASAP as it is a manufacturing defect.

IT IS VERY IMPORTANT THAT YOU HAVE EXTERNAL SPEAKERS HOOKED UP ACCORDING TO OWNERS MANUEL, CORRECT OHM SETTING, ETC. IF THAT DOESN'T CORRECT THE PROBLEM, CHECK THE TUBES THEMSELVES.

Check for a shorted tube in the power amp section and check the power supply caps for leakage, let me know your results.

This is a serious mistake and one starting point could be checking your power supply capacitors, certainly they will be out, thing is if this is the only components that got fried or the issue goes beyond and reaches power amp module, let me know furthermore.

Hard to say what device you have. Peavey makes a lot of things.

If this happened to be a peavey amp that has effects send/receive, a bad "send" jack could cause the problem due to the integral switch that sends the signal to the power amp.
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The preamp section has a problem. It could be anything in the preamp but is OFTEN the input jack itself. They take a lot of abuse.

Also the insert jack contacts that forward the preamp to the power amp if no external FX are used can go bad.

Additional things are circuit board cracks caused by rough handling and also controls that have had the knobs mashed in which can destroy the potentiometers they are connected to.