Watching a rented movie, it stopped then screen turned blank

Try searching for the name of one of your videos on the iPad's spotlight search screen - swipe your first homescreen to the right, type in it's name, and if it finds it tap on it to see if it plays.

If that doesn't work then try closing the Videos app completely : from the home screen (i.e.not with the Videos app 'open' on-screen) double-click the home button to bring up the taskbar, then press and hold any of the apps on the taskbar for a couple of seconds or so until they start shaking, then press the '-' in the top left of the Videos app to close it, and touch any part of the screen above the taskbar so as to stop the shaking and close the taskbar.

A third option if the two above don't work is a reset : press and hold both the sleep and home buttons for about 10 to 15 seconds (ignore the red slider), after which the Apple logo should appear - you won't lose any content, it's the iPad equivalent of a reboot.