You are looking at using SumIf used like this:
=SUMIF(B1:B24,"<="&D2,C1:C24)-SUMIF(B1:B24,"<"&D1,C1:C24)
where D2 is your latest date to look before and D1 is your earliest date to look after, the ranges i have used are for my entire range so i had entries between B1:B24 for dates and C1:C24 for things to count!