This saves your credentials on your computer so that you won't need to login on future visits.

You should NOT use this on computers that you share with other people.

Normally you will be auto-matically logged out after one hour of inactivity, but checking the Remember Me option pushes it to 12 hours.

In any situation, clicking on the Logout button will immediately log you out and also "forgets" your credentials (the opposite of "Remember Me").