Alright let's do this. Leeeeeeeeeroooooooy Jeeeeeeeeeenkiiiiiiiiiiiiiiins!
Sorry, couldn't pass that up :)
First, to answer your question about the onboard card. Take a look at this picture
http://attachments.techguy.org/attachments/188229d1298768706/monitor-sockets.jpg
Where it says "Installed Card" there's two connectors, a blue one and a white one. Compare that to the back of your system and see if you have either one (blue or white or both) where it says "Onboard" If you do, then you have an onboard card.
In that case, do the following
- Turn off computer
- Unplug power cable from back of system
- Open up the side pannel on the computer
- remove the gfx card (unscrew the bracket, there's usually a small plastic piece that you have to move and release near the base of PCI slot)
- put the cover back on
- plug the power back in, connect the monitor to the onboard card
- turn the computer on and you should be good to go
Or just google "how to remove a video card" from another system and you can watch videos
In most computers, the BIOS is smart enough to realize that you don't have a gfx card installed and to default to the onboard card. If that is not the case, then you will need to plug the video card back in, connect the monitor to that and boot into the BIOS (normally accessed by pressing the F2 key or the delete key during startup) Then within the BIOS you have to find and change the option that says use onboard or PCI video card to Onboard.
Again, google it if you're not too sure and look up images.
With regards to the problem itself. One of three things is happening. Either your gfx drivers are too old or there's a connection problem (cables or gfx card inside the computer) or the card itself is failing. I would do the following in order.
Download and install the latest drivers for your video card from the card manufacturer's website. If you don't know what gfx card you have download and run this program, it will help you identify it.
http://www.techpowerup.com/gpuz/
If that doesn't fix it, then time to clean
Get a can of compressed air. Remove the video card from the computer using the steps above. Use the can of compressed air to blow clean the fan and heat sink on the video card. Once it's as clean as you can get it, then plug it back in, connect the cables and try again.
If that solves your problem, great, it was a heat issue.
If that doesn't solve your problem, then we're most likely looking at a busted video card. The only real way to test is to swap it out for a known good card or using the onboard card (if you have one)
If neither is an option, then you're better off taking the computer down to a computer store and asking for help testing.