When dealing with finances there are generally two different types of payments: payment in advance and payment in arrears. Each payment type has its own unique purpose based on the general contract type and the specific terms of each individual contract.
Payments in arrears occur when the service or contract terms are carried out first and then the payment is made. A typical example of a product that is paid for in arrears is meal in a sit down restaurant. In a sit down restaurant you eat your meal and then when you are finished you pay. Conversely, in a typical fast food restaurant you pay in advance and then receive your food.
Occasionally, interest on a loan may be payable in arrears. In this case, you pay the principle first and then pay the interest at maturity of the loan. A mortgage typically works in the opposite way. You pay interest on the mortgage as you go, with a small part of your payment going toward principle. As the principle is reduced, the amount of the payment going toward interest decreases and the amount going toward principle increases.
Zero coupon bonds are one of the few financial instruments in which it is more common to see interest payable in arrears. Typical bonds make interest payments monthly, quarterly or, most often, semi-annually. Zero coupon bonds work entirely differently; they are sold at a discount and slowly over their life become worth more until they reach full value at maturity. This allows the company to borrow the money and not make any payments until the maturity date. The provisions of the bond usually require the company to set aside money to cover the bond payments over the life of the bond.
Other common occurrences of payments in arrears include transactions like electric bills and phone bills, which must be paid in arrears because the company doesn't know how much you will use until after you have used it. On the other hand, expenses like rent are usually paid in advance, and may even require additional payments like deposits and fees before your are able to move in.