It should keep you logged on even if you close the browser due to the cookie it drops when you log on. You can close the browser and go to the page a day later and it will still show you as logged in. When you log off the site using the log out at the bottom of the page you clear the cookie(s) then. You can clean the cookie(s) out by cleaning your browser history. It will make you log on then like you logged off the last time. But without logging out the site probably keeps you logged in. You just don't see it because of the cleaned cookie(s)
In a nut shell, Always log off the site using one of the log off links on the page. There's one under your name on the main page and on the bottom right of all the other pages on the site.