In economics, the term "elasticity" refers to how much the demand for (or supply of) a product changes when certain other variables change. For example, price elasticity of demand looks at the change in quantity demanded for a good or service when the price of that good or service changes. Arc elasticity is simply a slightly more sophisticated way of finding elasticity.
The simpler formulas for finding elasticity tend to take the percent change in demand (or supply) and divide it by the percent change in the independent variable (say price). These formulas tend to get the percent change in demand by subtracting the quantity demanded at one price from the quantity demanded at the second price.
However, this can yield different values even for the same change, depending on which way the change goes. (Please see the about.com link for an example. To remedy this, arc elasticity instead uses a slightly more complicated formula that uses the average of the old and new quantities demanded as the denominator for finding percent change.
Again, what this means is that arc elasticity is a more sophisticated and consistent measure of elasticity.
The average elasticity for discrete changes in two variables, X and Y. The distinguishing characteristic of arc elasticity is that percentage changes are calculated based on the average of the initial and ending values of each variable, rather than only initial values. Arc elasticity is generally calculated using the midpoint formula. For infinitesimally small changes in variables X and Y, arc elasticity is the same as point elasticity.