How is the interest on my credit card balance calculated each month?

Most issuers use the average daily balance method with daily compounding. They take your APR and divide it by 365 to get a daily periodic rate. Each day, that rate is applied to your balance, and the day's interest is added, so the next day's interest is charged on a slightly larger amount, which is compounding. At the end of the cycle, those daily charges add up to your interest.

Because it's a daily average, carrying a balance even a few extra days costs more, and making a payment mid-cycle reduces the average and therefore the interest. This is also why paying only the minimum barely dents the balance, since a big chunk of your payment covers accrued daily interest. The 'interest charge calculation' box on your statement shows the exact method and daily rate.