Arc elasticity is the average elasticity measured over a finite change between two points rather than at a single point.
The midpoint formula
For quantity (Q) and price (P), arc elasticity of demand is commonly written as:
$$ E = \frac{(Q_2-Q_1)/\left(\frac{Q_1+Q_2}{2}\right)}{(P_2-P_1)/\left(\frac{P_1+P_2}{2}\right)} $$
Using midpoints makes the measure less dependent on whether you calculate the change from the first point to the second or in reverse.
Why economists use it
Point elasticity is useful when you already have a smooth demand curve and want the elasticity at a precise point. Arc elasticity is more useful when data comes as before-and-after observations or discrete changes, such as a price increase from one month to the next.
Interpretation
The sign and magnitude are interpreted the usual way. For demand, a negative value reflects the inverse relationship between price and quantity demanded. A larger absolute value means quantity is more responsive to price.