As mentioned by searchfgold6789 you want to check the "Remember Me" or similar option so "the password" is saved.
Actually, the way this feature works on any website is by using a cookie. If you have checked those options and it still logs you out, that means Firefox is not saving the cookies between sessions. By default, Firefox is setup to not delete the cookies. However, you may have changed those settings and that means you lose "the password" each time you close Firefox and restart it.
Look at your Firefox settings you may find that the cookies get deleted. This is in the about:config page. Click on the button that appears and in the filter enter cookie to see what you get there. For example:
network.cookie.lifetime.days default integer 90
If that number is 1, then it would "forget" the cookie within 1 day.
Another reason why you may lose your session: if you use the same account from several computers, the website you are accessing may log you out of the older sessions. However, if that's the case you should see the same effect in all browsers and all OSes (i.e. log under Windows IE, then in Firefox, then back in IE you would be logged out.)