|If your dates are in Row 2, then I have to assume that your names actually begin in Row 3.|
I also assume that you have a matching list of names in Column A of Sheet2 and that you want the duties for the entered date to appear next to each name (Column B)
You don't say where you want to enter the date in Sheet2, so I'm going to assume A1.
In Sheet2!B3 enter this formula:
Drag it down to the bottom of your list of names.
This will use the MATCH function to return the column number within Sheet1!B2:AD2 where the date entered in Sheet2!A1 is found.
It will then use that value as the column offset form Column A and return the value in that cell.
For example, if 10/23/2010 is found in D2, that is the 3rd column of the B2:AD2 range and it will return the values in Column D.