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Try using the nomodeset or an alternative boot parameter.
From your Bios screen, press and hold down Shift key until you see the Grub menu. Highlighting the first entry there, press 'e' to edit it. Using arrow keys, navigate to words "quiet splash", delete them and type "nomodeset" in their place (without quotes). Press Ctrl + X to continue boot. Hopefully, you'll see the desktop this time.
Once on the desktop, make sure you are connected to the Internet. Now go to System > Administration > Hardware Drivers (Lucid) or Additional Drivers (Maverick) and check if any proprietary drivers are available for your graphics card. If there are any, activate the current/recommended one's and the problem should go away forever.
nomodeset alternatives for other Graphic Cards
According to our senior member oldfred at ubuntuforums.org, you need to replace nomodeset with some other parameters for different cards.
Older Intel video card: i915.modeset=1 or i915.modeset=0
Generic: xforcevesa or nouveau.modeset=0