According to a bug report, a recent Ubuntu 14.04 update broke the desktop for some Nvidia Optimus users:
"Until yesterday the 14.04 install on this Thinkpad T530 (nvidia Optimus) with nvidia-331 drivers worked just fine, including excellent multi-monitor behavior. Yesterday the set of packages below  was updated. I shut down, booted this morning and found out that I was not shown the lightdm login screen: I was stuck at the purple screen that precedes it (into which I enter my disk encryption password)".
- lp #1365695 bug
This bug doesn't affect all Nvidia Optimus users though. In fact, my laptop has Nvidia Optimus and this bug doesn't occur for me. That's why I can't guarantee that the solution below will work 100% however, I don't see why it wouldn't (and it was confirmed by users in the bug report comments).
Until the Ubuntu developers fix this Nvidia Optimus issue (the bug itself is caused by the ubuntu-drivers-common package), here's a work-around / temporary fix that you can use to get your Ubuntu desktop back. Press CTRL + ALT + F1 (because, if you're affected by this bug, you can't access the desktop), log in via the command line and type the following command:
sudo apt-get install ubuntu-drivers-common=1:0.2.91.4 nvidia-common=1:0.2.91.4
The command above downgrades the ubuntu-drivers-common and nvidia-common packages to the last working version available in the repositories (0.2.91.4).
Then reboot your system using:
Note that you must not upgrade to the latest nvidia-drivers-common 2.91.6 (or to 2.91.7 from the Ubuntu Proposed repositories) because both seem affected by this bug. Check the bug report to see when the bug is fixed and only then upgrade this package.
Thanks to Fabio Colella for the tip! Image via