In economics, elasticity is defined as the proportional change of a dependent variable divided by the proportional change of a related independent variable, at a given value of the independent variable. The concept was introduced by Alfred Marshall in the context of the law of supply and demand, and it is explained with great clarity in his Principles of Economics. 
For a mathematical explanation of the following concepts see the tutorials subpage.
Price elasticity of demand
The best-known application of the concept of elasticity is to the effect of a price change on the demand for a marketed product. The "price elasticity of demand" for a product is the proportionate decrease in demand for a product divided by the proportionate increase in its price. If the elasticity of demand is zero, demand is insensitive to price. If the elasticity of demand for a product is greater than 1, a price increase will lead to a fall in the amount spent on the product and if it is less than 1, a price rise will result in a rise in the amount spent on it.
The "cross-price elasticity of demand" between two products is the proportional change in the demand for one of the products divided by the proportional change in the price of the other. If the two products are substitutes such as ale and lager, the elasticity is positive, and if they are complementary goods such as cars and petrol, the elasticity is negative.
Income elasticity of demand
The "income elasticity of demand" for a product is the proportional change in the quantity of the product that consumers will buy divided by the proportional increase in their average income. Luxury products such as jewellery tend to have elasticities greater than 1, basic products such as milk tend to have positive elasticities less than one, and "inferior goods" such as second-hand clothes tend to have negative elasticities. Unless there is a change in the proportion of the community's income that goes into savings, its spending must rise in proportion to its income, and the average income elasticity of demand for all the products that it buys must therefore be 1.
The concept of elasticity is applicable to any relationship which can conveniently be summarised as the ratio of two rates of change. Its application to demand has parallels in the analysis of supply so that, for example, the price elasticity of supply is the proportional increase in the supply of a product resulting from a proportionate increase in its price. It also has applications to the analysis of the production function. For example, the "elasticity of substitution" between two inputs is the proportionate change in their relative use divided by the proportional change in their relative marginal products (which, in the case off pure competition is the same as the proportional change in their relative prices). For example, a high elasticity of substitution between labour and capital would motivate a firm to substitute capital for labour in response to an increase in wage rates.
Inferior goods are goods whose demand increases when income falls. This is because they provide a similar service to a more expensive good, but are considered less desirable. Examples include public transportation and thrift store merchandise.See also Supply and demand