Essentially, interest is nothing more than the cost someone pays for the use of someone else's money.
The interest rate that applies to investors is the Federal Reserve's federal funds rate. This is the cost that banks are charged for borrowing money from Federal Reserve banks. Why is this number so important? It is the way the Federal Reserve (the "Fed") attempts to control inflation. Inflation is caused by too much money chasing too few goods (or too much demand for too little supply), which causes prices to increase. By influencing the amount of money available for purchasing goods, the Fed can control inflation. Other countries' central banks do the same thing for the same reason.
Basically, by increasing the federal funds rate, the Fed attempts to lower the supply of money by making it more expensive to obtain.
Interest Rates and Investors
Investors can choose to invest in stocks or bonds, and their investment choice is based on the expected returns of each investment. For example, suppose that the Fed decides to lower interest rates. When interest rates are lowered, bonds are issued with a lower interest rate. As a result, investors will realize they can earn more money by investing in stocks, and they will do so. When more people choose to invest in stocks, stock prices will start to rise. Similarly, when the Fed decides to raise interest rates, bonds will be issued with a much higher interest rate. This will encourage investors to purchase bonds, and as investors sell their stocks, stock prices will fall.
Interest Rates and Consumers
When the Fed decides to lower interest rates, consumers are encouraged to purchase items such as houses and cars because they can finance these purchases at lower interest rates. As consumers start to spend, they increase the earnings of companies. As investors notice the earnings of a company start to increase, they will purchase shares in the company and this will cause the stock price to rise.