General: A process can be ignored adiabatically if it happens so fast compared to other processes (that are of more interest) that it can be assumed to reach an equilibrium instantly. This is known as the adiabatic approximation.
Examples: (1) When a gas expands adiabatically, there is no flow of heat energy into the gas. Thus, the gas cools as its internal heat energy is used to drive the expansion. (2) An adiabatic wall prevents the flow of heat through it. It is the theoretical idealization of the wall of a thermos flask.