Can they take my car away if I don't make a payment?

Yes they can, they will normally come in the middle of the night and tow it off ( unless they have a set of keys for it which happens on most buy here/pay here lots) or they let you drive to work and then take it from your parking area at work.

But yes they will send you a few notices of non payment and then they will come get it. If you don't pay to get it back, they will re-sell it and you will still owe the balance from what you owed to what they got paid plus costs to repo and sale the car. ** which is often as much as you owe now on the car due to added costs of the sale