Ubuntu gets stuck in a login loop
Did you end up here after running sudo startx? Nevertheless:
Press Ctrl+Alt+F3 and login into the shell.
Now run ls -lA. If you see the line
-rw------- 1 root root 53 Nov 29 10:19 .Xauthority
then you need to do chown username:username .Xauthority and try logging in (you may also need to do the same for for .ICEauthority).
Else, do ls -ld /tmp. Check for the first 10 letters in the left: they should read exactly so: drwxrwxrwt.
drwxrwxrwt 15 root root 4096 Nov 30 04:17 /tmp
Else, you need to do sudo chmod a+wt /tmp and check again.
If not both, I'd recommend you either
sudo dpkg-reconfigure lightdm- or uninstall, reinstall it.
Now press Alt+→ until you reach the login screen again, and restart.
I had this and after looking at /var/log/Xorg.0.log I found out that it's a Nvidia problem (there was a line saying Xlib: extension "GLX" missing on display ":0).
I realized I have Nvidia drivers from official website which are not really stable and tested (so I've read and also experienced in the past).
The solution here was to install package nvidia-current from Ubuntu repos; it is an awfully outdated version, but it's tested properly at least. Its installer is quite capable too and it uninstalled successfully my hack-installed unstable version from Nvidia website.
TL;DR, just try logging into the shell (Ctrl+Alt+F2 or whatever F between F1 and F6) and type
sudo add-apt-repository ppa:graphics-drivers/ppa
sudo apt update
sudo apt install nvidia-367
If it succeeds, reboot.
sudo reboot
If you're lucky enough, problem solved, you should be able to login to Unity.
UPDATE
Please note that sometimes nvidia-current might install the wrong driver. In that case, search the latest compatible driver for your video card and install it. For example, on Ubuntu 16.04, nvidia-current points to the version: 304.131-0ubuntu3. This might be incompatible with your graphics card; therefore, search with sudo apt-cache search nvidia-[0-9]+$ for the package you need, and install it.
I encountered this exact problem and none of the suggested fixes above worked for me. After almost giving up I looked at the .xsession-errors and noticed I had a typo in my .profile (I had an extra } in the file after I edited it earlier in the day).
That was causing the login loop. It might be another place to look if the other suggested fixes don't work for you.