The basics of interest
Interest is the cost of borrowing money. So, an interest rate of say 5% per year means the borrower promises to pay an extra 5% of the outstanding loan amount to the lender each year.
Simple interest
Like the name suggests, simple interest is the most basic kind. In our 5% per year interest example, on a $10,000 loan simple interest means the borrower would pay the lender an extra $500.
Compound interest
Compound interest is slightly more complicated and much more common. In easy terms, it's interest on interest. Compounding is done by a set period.
In our 5% per year interest example, on a $10,000 loan with annual compound interest, the borrower would owe $500 in interest at the end of the first year. The new balance of the loan would then be $10,500.
The 5% interest would then be applied to the new $10,500 balance so at the end of the second year, the borrower would owe $525 in interest that year. The interest grows each year (it "compounds") until the loan is paid off.