Most systems beep at least once on powerup. If there's no beep either
the internal speaker is disconnected or the computer has an almost
completely disablibing hardware problem. The problem could be a variety
of things including the motherboard, CPU, memory strips or power supply.
 Without ANY test equipment the best you can do would be to disconnect
everything BUT the motherboard, CPU, CPU fan, first memory strip, video
card (if present) and display for a test.

If the computer can operate well enought to operate the speaker, but not
generate a video signal most computers will generate additional beeps
that may give some indication of what is malfunctioning. You would need
a list from the motherboard manufacturer to determine what a particular
beep pattern means.

A simple multimeter wouldn't cost much at some electronics chain and
home repair centers. That would allow you to test the drive power leads
to determine if the power supply is operating. Each drive power lead
should have 12 and 5 volt readings relative to the black ground wire.
The power supply will NOT operate if there is a short connected to it OR
if there is no electrical load connected to its output.

If the hard drive light comes on and the fans are turning, but you hear no beeps and
get no signal whatsoever to the monitor, turn off and unplug the computer, open the
case, then remove and reseat the video card, making sure it is securely inserted all
the way into its slot.

If the problem still occurs after that, remove and reseat the memory (RAM) sticks.

If the fans and hard drive light do not come on when you start up, you have a power
supply, power switch, or motherboard problem.

The monitor, if it works on another computer, is not likely the problem.