Currently, Minecraft requires OpenGL 1.2 which nearly all graphics cards support. However, in 1.8 or 1.9 it will require OpenGL 2.0. This drops support for a great many graphics cards, unfortunately. What specific model do you have? This program will tell you what model you have and what OpenGL version it supports. http://www.ozone3d.net/gpu_caps_viewer/
The only fix I can think of is to update a driver or buy a new GPU (Graphics card).
http://www.iobit.com/driver-booster.php
This link has a program that I trust that will show any drivers to be updated.
If you need a new card, I'm afraid I can't quite help you there. Someone else or a quick Google search should help you find a card that will work.
Currently, Minecraft requires OpenGL 1.2 which nearly all graphics cards support. However, in 1.8 or 1.9 it will require OpenGL 2.0. This drops support for a great many graphics cards, unfortunately. What specific model do you have? This program will tell you what model you have and what OpenGL version it supports. http://www.ozone3d.net/gpu_caps_viewer/