Demand elasticity is a measure of how much the quantity demanded will change if another factor changes. One example is the price elasticity of demand; this measures how the quantity demanded changes with price. This is important for setting prices so as to maximize profit.
When price elasticity of demand is elastic, the firm should lower prices, since it will result in a big uptick in demand, increasing your total revenue. When price elasticity of demand is inelastic, you should increase prices because there will be only a small decrease in demand, and again, total revenue will increase. When price elasticity of demand is unit elastic, changing the price will not change total revenue, since price and quantity will generally change in lock step with each other.