Adiabatic process is a process in which there is no heat exchange between the surrounding and the gas.
So for example if an expansion happens really fast this process is adiabatic because there is no time for the heat to get transfered.
My professor said that we can describe this situation with ideal gas laws. But how can you do that while ideal gas law only hold in equilibrium states and if the process happens very fast there is no enough time for the pressure to become equilibrium pressure?
What we can calculate is if we make a box that is completely isolated and lower the pressure slowly.
It is useful and quite common to use the term 'adiathermal' to describe a process with no heat exchange, and 'adiabatic' to describe a reversible adiathermal process. If a gas is allowed to expand into a new volume by the removal of a barrier (perhaps what you had in mind), the process is irreversible and so by this definition it is not adiabatic.
In addition to the process being irreversible, during this process thermodynamic quantities such as pressure and temperature do not exist (are not well-defined). For this reason it would indeed make no sense to use the ideal gas law in the intermediate stages, since this relation presupposes the existence of these quantities.
I believe using the Ideal gas laws hold if you make a quasi-static approximation. I.e. you move through the expansion in infinite small steps, and say the gas remains in equilibrium in each step. Of course, it's an approximation, just as a truly adiabatic chamber would be.
I think speed of expansion doesn't matter here. You can have Adiabatic process happen in all the cases provided there is no heat exchange from outside the system. May be your teacher was giving you some example wherein fast natural expansion was the close approximation of an Adiabatic process. Remember, ideal gas is only an assumption, there is no real ideal gas exist. But most of the gases are very close approximation of an Ideal Gas.