Spidey:
The problem is your refresh rate most likely. Older monitors can't handle higher resolution settings due to this. From what you've described above this is whats happening is usually the cause of it in about 90% of cases when dealing with older hardware, not including production flaws in various components.
If you get picture and its fine on 800 X 600 then your video card is working fine and isn't having a conflict with your monitor, otherwise you'd get no picture at all, or at least one thats been scrambled in some fashion.
Now if your computer is saying that is has "Integrated Intel AGP graphics" then you may want to check inside the box or call the dealer to make sure your video "card" isn't built directly into the motherboard. If it is then damn. . . For the most part built in sound and video cards really kinda suck.

Not that intel stand alones are all that hot either.

How many years old is your monitor and does it have a maufacturer sticker or plate on the back or bottom. This should tell you the make and model number and then its much easier to find out what the exact problem is if it goes back several years.
I've built more then enough franken-puters to be familiar with this type or problem.