"Story cards" are an array of index cards used by the writers of Futurama to lay out the overview of a plot for an episode. After having laid out the basics for an episode, a writer can then be assigned to write it.
Some of these index cards are blue, thus called "blue cards" or just "the blue index cards", which contain all the often used plot devices in a Futurama episode. There is generally three blue cards; "trial", "run away" and "tour".
Due to their over usage, these blue cards have become worn with time.
The purpose of such cards is to find a way to end, begin or continue a plot. A "tour" is a good example on how to start an episode, while "trial" is a good way to continue a plot and "run away" is an example of ending a plot.