The annual percentage rate puts the whole cost of a loan into one yearly percentage, fees included. Two offers at the same interest rate can show different APRs because one adds an arrangement fee and the other does not, and that is the point of it: it lets you compare offers that charge in different ways. The CFPB describes it as the interest rate plus points, broker fees and the other charges you pay to get the loan.
The lender finds the one rate at which everything you repay, discounted back to the start, equals the money you actually receive. Borrow £10,000 over 36 months at 7% with a £300 fee taken from the advance. The instalment is £308.77, set on the full £10,000, but only £9,700 reaches you, and the rate that balances those flows is about 0.756% a month. The UK and the EU compound that monthly rate into a yearly one, the APRC, which here is 9.46%; see the FCA's definition and the Consumer Credit Directive. A US lender multiplies the monthly rate by 12 under Regulation Z, so the same loan shows 9.07%. Germany calls the figure the effektiver Jahreszins (PAngV section 6) and France the TAEG.
Neither figure is the rate your instalment is worked out from. That is the nominal interest rate in the agreement, 7% in the example, and putting the APR in its place overstates the payment. The car loan calculator therefore asks for the contractual rate and takes financed fees as a separate input, and the loan calculator wants the contractual rate too, with fees added on when you compare total costs. The credit card payoff calculator is the exception: a card's APR is the rate it charges, and the tool divides it by 12 to get each month's interest.