Odd… very odd.
What did you do before this happened? Install or uninstalling any software?
Reboot. Then when you reach the login screen press ctrl alt f3
This will switch to a tty
Login
Type the following carefully to reinstall the key packages
sudo apt install --reinstall lightdm slick-greeter budgie-desktop budgie-core gir1.2-budgie-1.0 libbudgie-plugin0 libbudgie-private0 libbudgietheme0 libraven0 budgie-control-center budgie-desktop-environment ubuntu-budgie-desktop
reboot --
Now try logging in as normal.