I'm a little surprised not to find this information in the faq, but anyway, it is described in a Stack Exchange blog post. In summary:
- Everything is seamless and I'm logged in automatically
If you visit a Stack Exchange site and you find yourself logged in right away, that means you are still logged in from your last visit to that same site. The "usr" cookie is valid for 6 months, and is refreshed every now and then when using the site.
- Or I get a big notification bar saying "Welcome back Mr Lister; Click here to refresh"
When you see this notification bar, it means that you either have never visited or have been logged out from this particular site, but you have recently logged into another site on the network. (I'm having a hard time finding out how recent is recent in this case, but I think it might be on the order of 30 days; hopefully someone will offer a number for that.) Note that it's not enough to just have visited another site in the SE network - you actually have to have gone through the authentication process within the time frame.
- Or I'm not logged in at all and I need to click the Log In link manually
This happens when you don't have an account on the site you're visiting, or when you do but you haven't authenticated recently enough on another site. Note that accounts are associated using the OpenID as a key, so if you have different accounts on different sites that use different OpenIDs, authenticating with one won't enable the automatic login for another.