My Aunt has a computer Windows 98 and for some reason, the computer comes on and you can hear the ram, HD starting but the monitor wont come on. I've checked with a good monitor and its the same thing.... What could this most likely be? Graphics Card gone bad? Power Supply? Teh HD is working so I ruled out Hard Drive Failure.

Any beeps on Post? If so, how many?
One beep is good. No beeps, or anything other than one, is bad

Whats the status of the power light on the monitor?
Does it turn on and then go yellow?

Try reseating the video card if not integrated and the memory.
Check all cables as well

No Beeps.
Monitor's light is Orange like the computer has been shut down and monitor left on. I opened it up and all cables are good. Unfortunely the graphics card is integrated. The Memory is fine. The computer will come on and it sounds like its loading Windows..... ( You know the sound it makes on the loading screen) Also Keyboard lights flash when it first comes on.

Get a PCI or AGP (If the motherboard has an AGP slot) graphics card.

TT75 makes a good point. If you have a local used computer store you may be able to pick one up for about $10. At least that way you can rule the video card out.

Yea I can probably get one, from a place where my dad is friends with the owner.