This will start from today and include 7 days forward:
=SUMPRODUCT(--(A1:A14 >= TODAY()),--(A1:A14 < TODAY()+8))
If you want to include 7 days back as well:
=SUMPRODUCT(--(A1:A14 >= TODAY()-7),--(A1:A14 < TODAY()+8))
Addendum:
sorry, I forgot to explain that in my example, A1:A14 contained the ranges of cells of dates to be examined for being within the alloted period.