When you invest money, you earn interest on your capital. The next year, you earn interest on both your original investment plus the interest from the year before...
Compound interest is the process of earning interest on interest that you've already been paid. For example, say you have £100 and you earn 5% interest per year. After one year, you'd have £105 (£100 1.05). After two years, you'd have £110.25 (£105 1.05). The alternative to compound interest is simple interest, where you only ever earn interest on the original amount you invest. With simple interest, you would have £105 after one year, just as before, but £110 (not £110.25) after two years. That seems a trivial difference, but the effect of compound interest mounts up over long periods: after 30 years, you'd have £250 using simple interest, but £432.19 using compound interest.
You can use the power of compound interest to good effect with dividend-paying shares. Instead of spending the dividend you receive, you use it to buy more shares in the company. This in turn gives you more dividends. Repeat this process for long enough and you can turn a small initial sum of money into a large one. This can be the case even if dividends per share or the share price stay the same.
Let's say you buy 1,000 shares in WidgetWorks at 100p per share, when it is paying an annual dividend per share of 4p. Over the next 30 years dividends stay put at 4p per share and the share price stays at 100p (an unlikely scenario, but useful to illustrate a point). If you had kept your 1,000 shares you would have received an annual dividend income of £40 (1,000 x 4p), or £1,200 over 30 years. With your 1,000 shares still worth £1000, your investment value would be £2,200. But if you had reinvested the dividends and bought extra shares each year, you'd end up with a much better result. At the end of 30 years you would own 3,243 shares worth £3,243 and have an annual dividend income of £125.
Watch Tim Bennett's video tutorial: Compound interest: the lazy way to get rich.