it is, in fact very important in a lot of situations to understand not only qualitatively but also quantitatively how responsive quantities such as demand and supply are to things like price, income, the prices of related goods, and so on. For example, when the price of gasoline increases by 1%, does the demand for gasoline go down by a little or a lot? Answering these sorts of questions is extremely important to economic and policy decision making, so economists have developed the concept of elasticity to measure the responsiveness of economic quantities.
Elasticity can take a number of different forms, depending on what cause and effect relationship economists are trying to measure. Price elasticity of demand, for example, measures the responsiveness of demand to changes in price. Price elasticity of supply, in contrast, measures the responsiveness of quantity supplied to changes in price. Income elasticity of demand measures the responsiveness of demand to changes in income, and so on. That said, let's use price elasticity of demand as a representative example in the discussion that follows.
Price elasticity of demand is calculated as the ratio of the relative change in quantity demanded to the relative change in price. Mathematically, price elasticity of demand is just the percent change in quantity demanded divided by the percent change in price. In this way, price elasticity of demand answers the question "what would be the percent change in quantity demanded in response to a 1 percent increase in price?" Notice that, because price and quantity demanded tend to move in opposite directions, price elasticity of demand usually ends up being a negative number. To make things simpler, economists will often represent price elasticity of demand as an absolute value. (In other words, price elasticity of demand could just be represented by the positive part of the elasticity number, eg. 3 rather than -3.) Conceptually, you can think of elasticity as an economic analogue to the literal concept of elasticity- in this analogy, the change in price is the force applied to a rubber band, and the change in quantity demanded is how much the rubber band stretches. If the rubber band is very elastic, the rubber band will stretch a lot, and it it's very inelastic, it won't stretch very much, and the same can be said for elastic and inelastic demand.
You may notice that this calculation seems similar, but not identical to, the slope of the demand curve (which also represents price versus quantity demanded). Because the demand curve is drawn with price on the vertical axis and quantity demanded on the horizontal axis, the slope of the demand curve represents the change in price divided by the change in quantity rather than the change in quantity divided by the change in price. In addition, the slope of the demand curve shows absolute changes in price and quantity whereas price elasticity of demand uses relative (i.e. percent) changes in price and quantity. There are two advantages to calculating elasticity using relative changes. First, percent changes don't have units attached to them, so it doesn't matter what currency is used for price when calculating elasticity. This means that elasticity comparisons are easy to make across different countries. Second, a one-dollar change in the price of an airplane versus the price of a book, for example, are likely not viewed as the same magnitude of change. Percentage changes are more comparable across different goods and services in many cases, so using percent changes to calculate elasticity makes it easier to compare the elasticities of different items.