They measure how much the chain stretches, and at a certain percentage of stretch they recommend you change it. If the chain and cassette are badly worn the chain will start skipping, depending a bit on how you pedal.

You can see if the cassette (or chainrings) are getting worn by looking at the teeth (are they called teeth in English? anyway) - they'll be sharp on top and the chain will have gouged on one side creating an assymetrical profile like waves, while new they'll be "mesalike"- flat on top and symmetrical sides.