An adiabatic process is a process in which no heat is transferred. This can happen if the process happens so quickly that there is no time to transfer heat, or if the system is very well insulated from its surroundings.
Q = 0
In that case the First Law states:
DEint = -W
If the system is allowed to expand, doing work, the energy comes from the internal energy - the gas must cool.
The P-V diagram for this process shows the system cooling as it expands.
It can be shown that the equation governing this path on the P-V diagram is:
PVg = constant