IMPORTANT NOTE: this information is subject to change until Blender 2.8’s final release.
Blender 2.8 is as much of a change from version 2.79 as 2.50 was from 2.49 in that significant parts of the internal rendering system has been rewritten to accommodate more modern systems and effects. Unfortunately this mean it may not run when installed to older computers, desktop or laptop alike, unless the graphics sub-systems support OpenGl 3.3 or above.
How to check Blender 2.8 compatibility.
To check if the “Unsupported graphics driver” error is the actual problem starting Blender 2.8, first determine the systems graphics capabilities.
If this is unknown, type “Device Manager” into the Start menu Search bar (or Cortana) and check the “Display adapters” listing, it should read something like “Intel(R) HD Graphics 3000” for example, that’s mid-second generation Intel CPU (2xxx).
Using this information search the Internet for details. For the above this might bring up this from Intel itself or this from Wikipedia, the upshot being that the HD 3000 only supports Open GL up to 3.1 on Windows. In this instance Blender 2.8 won’t run on this system resulting in the error seen.
How to fix the Blender 2.8 compatibility issue
Notwithstanding driver updates, unfortunately the only way to fix or address the issue and get Blender 2.8 up and running is to update the systems graphics hardware, the graphics card, onboard chipset or component responsible for rendering graphics, to something that’s OpenGL 3.3 compatible. For laptop/notebook users this is generally not possible. For desktop users any modern card manufactured within the last few years should be capable of running Blender 2.8 but double-check before purchase.