Besides being something your mama used to do when she went to the store, layaway is an arragement between you and a store owner where you buy something (ex: a suit) and pay cash a little at a time.  The store owner keeps the suit until you finish paying the item off (usually over a period of a few weeks or months).
Layaway used to be common when there weren't so many credit cards and store cards around, and is still a great way to buy things because you usually don't have to pay high interest rates on top of the purchase price of an item.  The bad news is that most major stores don't offer layaway anymore because they would rather have you use their store cards (so they can make all the money on those high interest rates).