I'm making a background image for a PDF document. I have just realized that the same jpeg background looks really different (different colors) on different screens. Is there a way to avoid this huge bias? Are there any suggestions?
There are measures you can take to work with web compatible color profiles as SteB mentioned, but the overal problem you're describing is generally an accepted downside to other people being able to view your document on their own screens.
Most computer screens even fluctuate ever so slightly over time in regards to color, so even if you had the same model of computer screen side by side you might be able to detect a difference.
So in short, there really isn't a way around this. It's not worthy losing sleep over though.
There's a free tool called Calibrize that will help you control your own color environment. But it still won't do anything for other's environments. Read the color management primer for an explanation of possibilities and pitfalls.
Calibrize is a free consumer grade software solution, and it sure is a heck of a lot better than nothing. But for professionals I'd recommend investing in a simple hardware measurement device such as a Colormunki Smile or a Spyder. Along with the right settings, this will make sure that at least your own system adhers to color standards. If it looks off on another monitor, it will be their fault, not yours. So it will only partly fix the color problem, but it will fully fix the issue of responsibility.
BTW, if you're on a Mac, you're in luck. Check your system settings> Monitors> Color. That should get you on your way.