There are several versions of this on the web, but I thought the following engineering scenario would provide some practical motivation.

You want to test a new cushioning material for automobile dashboards by placing a sheet of the material at the base of a 100-foot tower and dropping eggs onto it from various heights. Your goal is to determine the maximum height X--in whole numbers of feet--from which you can drop an egg and not have it smash to bits; and for fun, you have set yourself the task of accomplishing this with two eggs, in the smallest number of drops. (You may assume that the eggs are structurally identical, and that an egg which has survived a drop is unaffected and may be reused.)

Of course you can get lucky, start with a drop from 100 feet and have the egg survive, but that is leaving too much to chance. What you want to find is the smallest number N for which you can guarantee a determination of X in N drops. (And once you've solved that problem, find the smallest number of drops with which you can guarantee a determination of X with three eggs.)