This is because of the resolution of the game. If the game resolution doesn't match the normal resolution you use outside of the game, the resolution will change and cause strange effects on the other monitors. This works from left to right, so if you play the game on your right-most monitor, you will never experience this.
The solution is to play the game in the same resolution that you use normally for your monitor, or to run the game in a window so that the monitor resolution need not be changed.
It varies from game to game - some may cause problems, others won't. This is mainly caused by the game having taking over exclusive control of the video output, and not negotiating this correctly with the OS when switching in and out.
The standard mitigation is therefore to play in windowed mode. The game will still be within the OSes windowing system and so switching to other tasks is simple. This may incur a performance penalty, but that is mostly an artifact of older games - newer system libraries have minimised this.
This does however have the downside of not being fullscreen, which will detract from the experience.
If the game makes it availiable, an additional mode may be an option - usually referred to as windowed (borderless) or fullscreen (windowed), the game is rendered in a window without any window decoration. It can also be drawn over the taskbar. This provides the same experience as playing in fullscreen mode, but with the benefit of easilly being able to 'tab out' to other applications.
This also has the advantage of being able to play games fullscreen on one monitor, while using other applications on another, e.g. video player, web browser, which might not otherwise be possible in full screen mode.
The downside is that if it affects your game, you may get the same performance drop mentioned above.
Another alternative, also depending on the game supporting it, is to switch between fullscreen and windowed modes before and after switching to other applications.
The usual keyboard shortcut for this (when it exists) is Alt+Return.
While in windowed mode, switching to and from other applications works well, and when playing in fullscreen mode, there are no possible performance penalties due to playing in windowed mode.
Switching from fullscreen to windowed mode is usually handled better then switching to and from other applications, so does not usually result in the observed issues where the game is not rendered.
Best Answer
Perhaps not surprisingly, it depends entirely on the game(s). A solution that works for one game may not work for another, so you’ll have to see what works for whichever games you want to force windowed.
Some games natively support a windowed mode
-window)Running it in a virtual-machine (DOSBox, VirtualBox, VMWare, VirtualPC, etc.) may be a practical solution depending on the game and what kind of performance profile it has.
Another option is to run the game with a graphics emulator/wrapper that intercepts calls to DirectX/OpenGL/Glide/etc.
D3DWindower
Glide wrapper
DirectX OpenGL Wrapper
DXWnd (single-image window-forcing tutorial)