CTRL + ALT + F1 can be used to bring up a terminal. Once you log in press those keys and type in your username and password (the password will not display, just type it in anyway). Once that is done you will have a functioning terminal. Run the command
unity. Then press CTRL + ALT + F7 to return to the desktop.
If Unity is still not running then switch back to the terminal (with CTRL + ALT + F1) and see if there was an error, especially in the "Initializing" section, each of those should lines should end with "...done".
sudo apt-get install compizconfig-settings-manager followed by
export DISPLAY=0:0 followed by
ccsm then switch to the desktop, if it comes up make sure that the check box next to unity is checked then go to preferences on the left and click reset to defaults. After that restart the computer and try logging in.
Commands that might work:
This will add you to the xhosts, I doubt that you aren't in it but probably should try anyway.
Run this command to check if the computer supports unity, if it doesn't you can't run it. In that case I would click the Ubuntu logo on the login screen and select Ubuntu 2D and try that.
sudo rm /etc/X11/xorg.conf
This will reset your X11 config file.
If all of those fail you can reinstall unity:
sudo apt-get --purge remove unity
sudo apt-get install unity
sudo services lightdm restart
This will reset all of your graphical settings:
sudo killall -r -I gconf
sudo killall -r -I dconf
rm -rf .compiz* .gconf* .config/dconf/ .config/compiz*
If that fails this will reinstall the graphical portion of Ubuntu.
sudo apt-get install --reinstall ubuntu-desktop
You can try the same thing with compiz:
sudo apt-get install --reinstall compiz