I hope mtsu doesn't ridicule me for this one: HA
Adiabatic is really important when it comes down to undestanding meterology. Adiabatic spefically means the NO heat is exchanged with the surrounds of the gas or liquid. Yes, air cools as it rises but this is due to the amount of expansion or compression that parcel of air is allowed in that gain in altitude. Always think in terms of that and it will help; if a gas expands, it cools. Conversely, if it compresses it is heated. All this is due to how tightly knit the molecules are.
If you were to take a parcel of air and lift it will expand at nearly the same rate the atmosphere expands. If you were to measure the temperture along the way you would get its lapse rate. I won't go further unless it is warranted, andI am about to be late for the muuvvvie.