APR (Annual Percentage Rate) is mainly a way of measuring the cost of the loan you take. It takes the costs for the loan into consideration so if the loan is a no-fee loan APR should be same with the interest rate, otherwise a little higher according to the fees. This way you can understand the overall picture. Need to mention that some complex loan types (like mortgage for example) there might be other factors makes APR less reliable comparison tool. Also dont confuse with APY, which is use for the interest paid by banks for your deposit accounts. (Annual Percentage Yield)
As mentioned above without fees APR is same as interest rate. With the fees there is the formula:
Here r is the interest rate and n is the loan term. Add 1 to interest rate and loan term ratio and take nth power of this ratio at last subtract 1. It might seem a little complex but actually you can just try different values with the calculator above and see the affect of cost and rates to the APR yourself. Maybe some examples make it more clear.
Without fee the loan of 10000 with 12 months term you take with 5% interest rate Annual Percentage Rate is still 5. Now think that there was 300$ fee for application, broker, closing, insurance etc. Now APR is 10.56 as we add the costs to the payments and calculated the rate with total money we will pay. If you increase the loan term to 2 years APR drops to 7.91. This is because the fees spread more to the monthly payments. This also affects the early payment scenarios. If you are thinking closing the loan earlier APR might be underestimating the overall cost.