Price elasticity of demand is a measure of how much the quantity demanded of a good responds to a change in the price of that good. An elastic variable is one which responds disproportionately to changes in other variables. Similarly, an inelastic variable is one which changes less than proportionately in response to changes in other variables.
In more technical terms, it is the ratio of the percentage change in one variable to the percentage change in another variable, when the latter variable has a causal influence on the former. It is a tool for measuring the responsiveness of a variable, or of the function that determines it, to changes in causative variables in a unitless way.