The interest rate is the yearly price charged by a lender to a borrower in order for the borrower to obtain a loan. This is usually expressed as a percentage of the total amount loaned.
A more thorough definition of an interest rate can be found in The Economist's Dictionary of Economics. In part they define the "rate of interest" as:
The proportion of a sum of money that is paid over a specified period of time in payment for its loan. It is the price a borrower has to pay to enjoy the use of cash which he does not own, and the return a lender enjoys for deffering his consumption or parting with liquidity.
The interest entry by Paul Heyne at The Library of Economics and Liberty expands on this idea of the interest rate as a price which is determined by market forces:
The interest rate is determined by demand and supply: the demand for present control of resources by those who do not have it, and the supply from those who do have control and are willing to surrender it for a price. The question of exactly why demand and supply yield a positive rate of interest is one of the most fiercely disputed questions in the history of economic theory. It is enough to point out that when an individual acquires present command of resources, his or her set of available opportunities expands. In short, the present command of resources is something that people want. Therefore, those who get it are willing to pay for it, and those who give it up insist that they be compensated for doing so.
Note that when people discuss interest rates, they're generally talking about nominal interest rates. A nominal variable, such as a nominal interest rate, is one where the effects of inflation have not been accounted for. Changes in the nominal interest rate often move with changes in the inflation rate, as lenders not only have to be compensated for delaying their consumption, they also must be compensated for the fact that a dollar will not buy as much a year from now as it does today. Real interest rates are interest rates where inflation has been accounted for.