DISPLAY is used by the X Window System to specify the host, graphics controller, and physical display. The default :0.0 means local host (frequently a Unix domain socket instead of a TCP/IP socket), first graphics controller and first physical display attached (0 for both). If you were connecting to an X server on a different host, you would specify that host's name or IP address; if you were using a different controller or display you would specify that number.
XAUTHORITY is used by xauth (as mentioned by c4747p) and contains the name of the file with your X privilege tokens.
You are probably getting the message because a bad XAUTHORITY file leads one to have no permissions to attach to the X server.