This should work, but there is a little trick. After you enter the formula, you need to hold down Ctrl+Shift while you press Enter. When you do, you'll see that the formula bar has curly-braces around your formula. This is called an array formula.
For example, if the Months are in cells A2:A100
and the amounts are in cells B2:B100
, your formula would look like {=SUM(If(A2:A100="January",B2:B100))}
. You don't actually type the curly-braces though.
You could also do something like =SUM((A2:A100="January")*B2:B100)
. You'd still need to use the trick to get it to work correctly.