What is the difference between saturated and unsaturated adiabatic lapse rate?
The difference between a unsaturated and a saturated adiabatic lapse rate is that in the latter the water vapour has to change state from gaseous to liquid.
When air rises it expands due to the lower pressure of the ambient air at higher altitude. By expanding, the air parcel is doing work on its surroundings. Since no heat is transferred to the air parcel (by definition of being adiabatic expansion) but work is being performed, the temperature has to drop. Thus a rising parcel of air cools down.
Cooler air can hold less water vapour than warmer air. If the air parcel contains water vapour, at some point during the cooling the maximum relative humidity is reached. At that point the air is saturated with water vapour. If the air continues to cool, the water vapour content has to change from vapour to liquid. This is called saturated adiabatic cooling.
The process of condensation of the water content is exothermic; when the state changes from gaseous to liquid, heat is released to the air in the parcel. Therefore the lapse rate (temperature change with altitude) of a saturated adiabatic process is less than an unsaturated adiabatic process. The heat released from the condensation process reduces the lapse rate.