Compound interest is interest paid on interest. Each time interest is credited it joins the balance, and the next period's interest is worked out on the larger sum. Simple interest, by contrast, is always calculated on the original amount, so it adds the same sum every year.
Over a few years the two barely differ; over decades the gap dominates. £1,000 at 5% simple interest earns £50 a year and stands at £1,500 after ten years. Compounded yearly it reaches £1,000 × 1.05^10 = £1,628.89, and the extra £128.89 is interest earned on earlier interest. Leave it for thirty years and simple interest gives £2,500 while compounding gives £4,321.94. The general formula is FV = P × (1 + r ÷ m)^(m × t), where P is the starting sum, r the yearly rate, m the number of times a year interest is added and t the number of years. The SEC's Investor.gov runs the same arithmetic in its own calculator.
How often interest is added matters far less than the rate and the time. At 5%, moving from yearly to monthly compounding lifts the yearly yield to 5.116%, about an eighth of a point, while one more point on the rate adds a full point. Banks fold the frequency into a single figure, the AER or APY, which is the one to compare.
Debt compounds the same way. Card interest you do not pay is added to the balance and charged interest the following month, which is why a balance paid at the minimum shrinks so slowly. The compound interest calculator builds a year by year table of starting balance, deposits and interest, and the savings goal calculator runs the same formula backwards to find the monthly deposit a target needs.