I wish I had seen your question before. First of all, have a look at supported models here: Linux Download Center
If your card is supported, you can follow what I did below.
1) In GRUB menu, when the "Ubuntu" row is highlighted, you press "E" on your keyboard. It will open some booting command.
2) Look for "quiet splash". If you saw the word "nomodeset" right before that "quiet splash", just delete all that "nomodeset".
3) Press F10 to boot Ubuntu. Hopefully you can see your desktop.