This tutorial assumes you are using Microsoft Office Outlook 2007.  The same settings exist in Thunderbird, you will need to find them though.

in Outlook, go to the "Tools" menu and select 'Account Settings..."  The Account Settings dialog will appear.  You need to select the email account you wish to have leave the mail on the server.  Click on "Change".

The Change E-mail Account dialog will appear.  Then click on "More Settings...".  The Internet E-mail Settings dialog will appear.  Tick the box which says "Leave a copy of messages on the server" and set the time you want it to remain or alternatively untick it to have it remain forever (ONLY do this if you have LOTS of space!!)