If you click "Stay logged in" you... stay logged in.
I haven't had to log back into this site on my PC in... years?
Most of the time I check this site during free moments at work, on my work computer, so I don't want to stay logged in. Plus I just don't like cookies, etc. on my computer so I have my computers set to delete them anytime I disconnect from the internet. I seem to recall that somewhere I have a setting on this site to automatically sign-out after 30 minutes, but I'm not sure where that setting is; however I would think that should only apply if there was no activity, which isn't the case when it's happened to me.
EDIT: Ok, I found it. When I login there is a box checked for how many minutes I want to stay logged in, which I have set at 60 minutes, which should be more than enough time for me to spend here. Otherwise as Spider indicated there is another box I could check which says stay logged in.