Compound interest is when the interest earns interest. If you invest in something with compound interest, you earn not only the original investment amount but also any interest paid on the investment plus additional compounded interest that increases your total earnings. You can also lose money to compound interest if you are borrowing at a high rate of return. The term "compound" means that each time it accrues, it is applied to the principal and interest, resulting in exponentially increasing sums over time.
The term "compound interest" refers to the interest you would be paid on your investments that is added into your investment and then earn more interest. The reason that compound is so important is that it causes an exponential growth of wealth, with time, as opposed to just a linear growth. For example, if you invested $500 each year and received an 8% return, after 10 years you would have $8323.
In a nutshell, compound interest is the accumulation of interest on top of interest. Interest may be calculated to be paid at a certain rate annually or monthly.