Placing a tick in the 'Remember Me' box, when logging on, will cause your username and password to be stored in a file, known as a cookie, on your computer. This will mean that the next time you visit the site you will not have to enter your username and password because they will be read from this file.
Your username and password are encrypted in this file so other people will not be able to read them from the file. However, if you log on to this website using a computer that is shared by other people you should not tick this box. If you do then other people who use the computer would be automatically logged on as you.
The file that stores your logon details will be deleted if you click the 'Log off' link when using the site. Also if you delete your cookies from your Internet browser this file will be deleted.