Mathematically, interest generally falls in one of the following two categories:
Simple interest is of little interest to anyone because of inconsistencies inherent in the method.
When interest is added to a debt, the amount is the interest rate for the conversion period, times the amount of the debt.
When payments are made against a debt, the payments are credited to interest first, and to principal (outstanding debt) only to the extent that the payment exceeds the amount of interest due at the time of the payment.
When interest is not collected as it is accrued (as with a certificate of deposit, where the payment is in a lump sum), the interest increases the amount of money subject to interest. This is why simple interest produces mathematically inconsistent results.
Interest is sometimes referred to as rent on money. As with any rental, the market price (in this case, rate) is subject to change to reflect market conditions. Interest rates are very closely watched market indicators, and have a dramatic impact on finance and economics.
Interest involves the future, which is uncertain. Some interest bearing investments are riskier than others, and different parties will be offered different rates on loans. The measure of creditworthiness of an individual is called a credit rating or credit score. Other entities acquire a bond rating or credit rating by some other name.