Compound interest is interest calculated on both the original amount you put in and on the interest that's already accumulated. That second part is the key: your money starts earning returns on its own returns, not just on your original contribution.
Here's why that matters over time. Suppose you invest a lump sum and earn a steady annual return. In year one, you earn interest only on your principal. In year two, you earn interest on the principal plus year one's interest. By year twenty or thirty, the interest you're earning on past interest can dwarf the interest you're earning on your original contribution. The growth curve isn't a straight line, it bends upward.
This is why starting early matters so much more than the size of the contribution. Money invested in your 20s has decades for this snowball effect to work, so even modest contributions can grow substantially. The same dollar amount invested in your 40s has much less time to compound, so it takes a larger contribution to reach the same end result.
It also explains why compounding works against you with debt. Credit card balances that carry interest month to month compound the same way, just in reverse, which is part of why high-interest debt grows so fast if it isn't paid down.