Machines are said to be out of order(elevators, ATM machines, milling machines, etc). A car can also be out of order though in general conversation we say "broken down".
Out of commission means basically the same thing. But do bear in mind it often refers to factories or large things like dams. Those things when they begin operating, are said to be commissioned or brought on stream. So, if they are not in operation they can be said to be "out of commission".
People do sometimes say something is out of commission in every day language when they could have also said out of order. However, a large thing like a factory cannot be "out of order", technically speaking. It would be "out of commission".