I was receiving this error when I tried to start the Angry Birds game on Windows 7 Ultimate x64 and x86 versions. I don't know about other versions of Windows OSes.
I don't exactly know what could be the problem but from the error it is clear that the problem is related with graphics. As Windows 7 versions are mostly automatically install the required drivers for hardware, most of people do not install the graphics and other drivers. That could be one problem. But from some sources on internet, I came to know that Angry Bird is using OpenGL for graphics instead of DirectX; and OpenGL is not pre-configured/installed with Windows 7 installation.
I have a graphics card: nVidia GeForce 9400 GT (1 GB Graphics RAM and 4 GB DDR2 RAM, and LG W1953T 18.5in LCD monitor), and I had installed the drivers from the CD came with the graphics card and monitor. But still I was receiving the error. So, I thought installing OpenGL would work, but didn't found any setup of OpenGL on the internet. On some forums I had also found that downloading and installing latest graphics drivers will solve the problem.
If you have graphics card then go to the website of the graphics card's vendor.
If your motherboard has on-board graphics, then go to the website of motherboard's manufacturer.
You may also need to update the driver of your monitor.