Compound interest is when you earn interest on the money you previously earned in interest, when that money is added to your initial deposit. For example if you deposit a $1000 in a bank paying a 5% annual interest rate and left it there for 10 years, when you go to collect your money you would have $1629. Your money would have grown by more than $600 because of compound interest.
This occurs if the bank is compounding your money. For example if you earned $50 in interest on your initial deposit of $1000. The second year your earned $102.50 on your new balance of $1050. The third year you earned $157.63 on the current balance of $1102.50. This continues year over year. If you choose to leave the money there for another 10 years (20 total) your initial $1000 will grow to $2653 (more than doubling). Patience pays off with compound interest.