Money isn't free. For one thing, you have to pay to borrow money, so it has a direct financial cost. For another, when you tie up money in an investment, you can't use that money for anything else, so there's also an opportunity cost. Interest rates and discount rates both relate to the cost of money, although in different ways.
TL;DR (Too Long; Didn't Read)
Interest rates and discount rates both relate to the cost of money, although in different ways. An interest rate is the rate you can expect to pay for borrowing money, or the rate of return you expect from an investment. Discount rate refers to the rate used to determine the present value of cash.
Interest is the price a borrower pays to use someone else's money. Say you take out a $150,000 mortgage at a 6 percent annual interest rate. The bank didn't really "give" you $150,000. It's just letting you use its money for a while (up to 30 years). You'll pay the money back, of course, but each year, you'll also pay the bank 6 percent of your outstanding mortgage balance for the privilege of using its money. Car loans, credit cards and student loans all work on the same principle. Buy a bond or put money in a savings account, and you'll be the one earning interest: Someone will pay you for the privilege of using your money.
Interest rates reflect risk. The greater the risk that a loan won't be repaid, the higher the interest rate the borrower will have to pay. That's why people, companies and governments with poor credit have higher borrowing costs than those with good credit. But even "risk-free" loans will involve interest. U.S. Treasury securities, which the financial world generally views as having zero risk of default, still pay interest, albeit at relatively low rates. Investors expect to earn a return for their money, even if they put it somewhere safe.
In finance, there are two different things that go by the name discount rate. One is the rate that the Federal Reserve charges banks for short-term loans. The second definition is of more interest to investors – it's the rate you use when adjusting for the "time value of money." The time value of money is a basic principle of finance. It means that a certain amount of money has different values at different points in time. Given a choice between receiving $100 today and getting $100 in a year, you should take the money now. You could invest it, and if you earned any return at all (even a risk-free rate), you'd end up with more than $100 a year from now.
If $100 today is worth more than $100 next year, then there must be some value today that's equal to $100 a year from now. That value is called the "present value" of $100 a year from now, and you determine present value using the discount rate. If the discount rate is 10 percent, for example, then the present value is $90.00. If you invested $90.00 today and earned a 10 percent return, you'd have $100 a year from now. The trick, though, is in determining the proper discount rate. There are financial professionals whose entire jobs involve figuring this out.
An array of factors go into determining the appropriate discount rate to use in a time value of money calculation. For example, say an investment promised to pay $100 in a year. How much you'd be willing to pay for that investment today depends on the return you require to make it worth your while. Interest rates are one factor: You'll expect to earn a rate equal to your risk (and certainly better than the risk-free rate). Inflation is another: You want to make sure you don't lose ground while your money is tied up. Taxes also play a role: If you're going to take a tax hit on your profit, then that profit had better be worth it. And the return offered by similar investments will also factor in. If you can get a better return somewhere else, you might not bother with this one.