You may sometimes see repeated login attempts while logged in.  The
console window pops up, and a long string of attempted logins for common
names scrolls by.

This is a fairly standard attack in which a would-be intruder runs a
script that attempts to log in with common passwords and common account
names.  As these are all failed attempts, there is nothing to worry about
unless your passwords are weak and the login attempt succeeds.  If you
have chosen strong passwords, the script will be unable to log in to your
machine through brute force.

If the console window is a nuisance, you can move it aside so that it's no longer 
entirely on the screen.