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.
Best Answer
Some steps you can take to fix the login loop:
Go to terminal from login screen: Ctl Alt F1
login with user-name and password.
Run updates "sudo apt-get update"
Check all applications "sudo apt-get upgrade"
Check all packages "sudo apt-get -f install"
Update grub "sudo update-grub"
Remove all old applications and packages "sudo apt-get autoremove"
Reboot "sudo reboot"