I will guide you to the solution:
First Step
is to set the drop down menu's (combo box) on the range you want. In my example it will be in sheet1, having range A1:A10
. From menu Data, choose this option:

Set the criteria range, to be the range in sheet2 (values in combo box):

Now the values will appear in the first sheet:

Second Step
is to add the following formula in cell B1
of sheet1:
=ARRAYFORMULA(IF(ISBLANK(A:A)=TRUE,"";VLOOKUP(A:A,Sheet2!A:B,
{2} * SIGN(ROW(A:A)),0)))
This will automatically populate column B, with the values found in column B of sheet2, corresponding to the input given in column A of sheet1.
See the example file I've prepared: COMBOBOX and VLOOKUP