Dead motherboard or dead power supply?
 

I have an old computer that turns on (fans spin, hard drives spin, speakers beep) but there's no signal to the monitor.

Which one does it sound like, dead motherboard or dead power supply? Or is there ways I can test for it?

I wished I have an extra power supply around, but unfortunately I don't. I've also tried several different monitors and none of them worked.

Integrated video? Most likely motherboard.

Umm if it were p/s or mobo then nothing would turn on. Are you running on board video?
Try adding a 2nd party video card or vice versa.. Does your system even post bios?

could be m/b, ram, video..
if the psu has spikes, and unable to send/hold a proper voltage.. it could have that symptom.

i would start a process of elimination, try the psu on a different unit or try another psu.

speakers beep once as normal beeping or more than once?

did you connect your cable to your monitor recently? bent pins?

its likely a dead graphics card, sounds VERY similar to my computer at work after a power outage.

Count the beeps. Usually means something.

maybe its your ram if your computer beeps.. and the # of beeps have different meanings