Q Why, when I start an application, does it insist on re-opening all the files that I had open when I quit, even though I have unchecked the box to control that?
A This is supposed to be a good feature in versions of OS X since Lion, and should be controlled effectively by a checkbox towards the foot of the General pane in System Preferences. Older versions of OS X offer to ‘Restore windows when quitting and re-opening apps’, but its sense has now been reversed, to read ‘Close windows when quitting an app’.
It is possible that you did not notice this reversal of meaning: if you want each app to open clean, without the same windows as when it was last used, that checkbox used to be left unchecked, but now it should be checked.
It is quite possible that the preference list (.plist) file containing that setting has become munged, and needs to be trashed: this should be ~/Library/Preferences/com.apple.finder.plist, although the settings files containing the list of windows to reopen are saved for individual applications.
Updated from the original, which was first published in MacUser volume 28 issue 17, 2012.