Ticking the "Remember me" button when you log in will keep you logged in, on this computer, using this browser, for 60 days, even if you close the browser or turn the computer off.
Each time you visit within that period, the time is reset, so as long as you never leave more than 60 days between visits you need never log in again.
In the case of sites which can be viewed using more than one domain name, the persistent log-in will only work if you revisit using the same domain name. This is a security feature of your browser: for its purposes, even the presence or absence of the "www" will make a difference so, for instance, www.mysite.com counts as a different domain to mysite.com.
Using the "Log out" button at any point will clear this persistent log-in; in this case you'll need to log in again next time you visit.
This system relies on a cookie being stored on your computer: all it contains is enough information to verify your log-in when you return to the site.
Fill in your e-mail address, click the 'Send me a new password' button, and a new password will be sent to your e-mail address.