87 Corolla Black Box under hood is buzzing

I had a dead battery so I put a friends battery in started the car the with it running took it out and put mine back in. There is a black box under the hood with a few wires coming out of it and a throttle wire, it is now buzzing, buzzing buzzing, I dont know what it is. When i turn my key it starts then it also buzzes when the car is running.

You might have fried some/all of the ECU. That's why a clutch start of always better.

Yep, your ECU is now a paperwieght. Good thing, though, your car is older, you might be able to get away with it for awhile until you get another one from a junk yard.