The amount that a lender charges to a borrower for the loan of an asset, usually expressed as a percentage of the amount borrowed. That percentage usually refers to the amount being paid each year (known as annual percentage rate, or APR) but can be used to express payments on a more or less regular basis.
Interest rates can be either simple or compound. Simple interest is derived just from the original loan (known as the principal), but compound interest is calculated from the principal plus any interest accrued over the length of the loan. So if a $100 loan has 5% compound interest, then after one year the interest would be 5% of $105 (the original $100 plus $5 in accrued interest).
Most bank interest rates are derived from the base rate set by their central bank: the rate at which private banks can borrow from their central bank. Central banks use interest rates to control inflation and spending; by raising interest rates, the cost of borrowing and benefit from saving are both increased, so spending is discouraged. After the recession, many central banks dropped interest rates to encourage more spending.
Changes in the base rate can move markets in a major way, and so are a major event for traders. Traders can also speculate on changes in the interest rate, either via instruments like bonds or derivatives.