Assume dates in column A, Identity in column B and totals in column C. You can use SUMPRODUCT:
=SUMPRODUCT(($A$1:$A$20>=5/1/2007)*($A$1:$A$20<=5/31/2007)*($B$1:$B$20="Food"),$C$1:$C$20)
Adjust ranges as needed. Also you may need to work with date formats. It might be easier to set up start and end dates in two cells (i.e. D1 is start date, then D2 is End Date), and category in D3, then your formula would be:
=SUMPRODUCT(($A$1:$A$20>=$D$1)*($A$1:$A$20<=$D$2)* ($B$1:$B$20=$D$3),$C$1:$C$20)