How do I calculate monthly payments given a certain loan amount, APR, and loan term in years?

Using the following variables

s = principal
r = periodic rate
n = number of periods
d = periodic payment

Standard loan equation - formula derived by induction

d = r s (1/((1 + r)^n - 1) + 1)

If your APR is quoted in the USA it will be a nominal rate. For a loan with monthly repayments and a "nominal APR compounded monthly" the periodic rate r = APR/12

E.g.

APR = 5% = 0.05
r = 0.05/12 = 0.00416667

If your APR is quoted in Europe it will be an effective annual rate. To convert to a periodic monthly rate r = (1 + APR/100)^(1/12) - 1

E.g.

APR = 5% = 0.05
r = (1 + 0.05)^(1/12) - 1 = 0.00407412

See Investopedia - Present Value of an Ordinary Annuity for more insight into how this is calculated.

For example, the Investopedia loan illustration, with 5 annual payments and APR of 5% (nominal compounded annually is the same as effective annual rate.)

Using the same example of five $1,000 payments made over a period of five years, here is how a present value calculation would look. It shows that $4,329.58, invested at 5% interest, would be sufficient to produce those five $1,000 payments.

s = 4329.58
r = 0.05
n = 5

d = r s (1/((1 + r)^n - 1) + 1) = 1000