Gamma is a second order Greek letter and it measures the change in Delta by a $1 change in the value of the underline security (i.e. stock). Its importance is high because it determines how fast Delta is changing and we know that the latter is affecting greatly the premium.
Gamma is to Delta what acceleration is to speed. Suppose we have a call with premium $2, Delta 0.60 and Gamma 0.05. What will happen if the price of the underline security rise by $1? The Delta will change to 0.65 because Gamma is 0.05 and the premium will rise theoretically from $2 to $2.65, given that all the other factors which affect the option price remain constant. If we had a put with premium $3, Delta 0.60 and Gamma 0.05 then a $1 rise in the underline would have the same effect on Delta (from 0.60 to 0.65), but the put would have lost value, from $3 to $2.35.
Mathematically, Gamma is expressed by the first derivative of the option Delta relatively to a $1 change in the underline security and its formula is:
Gamma takes its highest value when the option is ATM and decreases when the option is moving towards ITM or OTM. Practically this means that Delta is more sensitive to changes in the underline security when the option is ATM and less sensitive when the option is ITM or OTM.