[excel] How to compare two columns in Excel (from different sheets) and copy values from a corresponding column if the first two columns match?

Make a truth table and use SUMPRODUCT to get the values. Copy this into cell B1 on Sheet2 and copy down as far as you need:
=SUMPRODUCT(--($A1 = Sheet1!$A:$A), Sheet1!$B:$B)
the part that creates the truth table is:
--($A1 = Sheet1!$A:$A)
This returns an array of 0's and 1's. 1 when the values match and a 0 when they don't. Then the comma after that will basically do what I call "funny" matrix multiplication and will return the result. I may have misunderstood your question though, are there duplicate values in Column A of Sheet1?