The dead zone on a controller is a small area around an analog stick's centre position in which the stick will not send any information to the console. It's kind of hard to explain, so I'll try to give an example of practical use.
Imagine you are playing a difficult platforming game. Your goal is to walk across a very narrow plank from left to right across the screen. You hold right on the analog stick- but you also bump the stick a little bit upwards as well. Because of the dead zone, small values for movement in any direction will be disregarded. Because of this, the small upwards movement is disregarded and your character runs straight to the right.
Generally it's used to make it easier to control things. Too small a dead zone and things feel twitchy, too high a dead zone and things feel unresponsive.
Hope that helped. :-)
Long-time love of video games