Most credit card issuers use the prime rate, which is based on the federal funds rate set by the Federal Reserve rate, as a starting point for their interest rates. The prime rate is generally what commercial banks charge their best customers.
For credit card customers, issuers take the prime rate and add an additional interest charge on top of it. The percentage used for that second layer can depend on the customer and may be related to factors such as:
Credit score
Income
Economic conditions
Those with higher credit scores typically have access to the lowest interest rates since the card issuer feels more confident that they will be paid any balance owed. Meanwhile, companies may see those with poor credit scores as being more likely to default on their accounts; therefore, they charge a higher interest rate to offset that risk.
Credit card interest rates that are based on the prime rate are variable and change whenever the prime rate changes. Banks can also adjust that amount they charge on top of the prime rate, so you could see a rate change even if the Fed doesn't make a rate adjustment. Although less common, a few cards offer fixed rates.