Just as you pay a person who works for you, say the neighborhood kid who mows your lawn, when a person lends you money that you can put to your own benefit, you need to pay that person for that. This payment (above and beyond the repayment of the loan's principal) is interest.
People receive interest when they loan out their money to others, such as depositing their cash in a savings account or other savings vehicle. The banks borrow that money and lend it out to others, paying you some interest in compensation.
The interest rate is the amount of extra money due, expressed as a percentage of the principal. So, if you receive 3% interest on your savings account, the bank is paying you $3 for every $100 in your savings account. If you pay 15% interest on your credit cards, then you pay Visa or Master Card or a similar company $15 for every $100 they lend you to buy things like groceries, dinner out, clothes, etc.
Interest rates change as the markets change. If money is tight throughout the system, lenders can increase interest rates because the supply is less than the demand from borrowers. Institutional lenders also might have to raise the rates that they charge borrowers if they themselves are being charged more in interest to get the money they work with.