It's accusative from Захар, which is a more common and modern version of this name (Захария is more archaic/religious). Захар is a full form, i.e. it's what's written down in a passport (eg. Захар Сергеевич Иванов). It has no short form per se, though some endearments (Захарка, Захарушка, Захарчик, etc.) can be used, and young people often shorten it to Зак.

This name is not very widespread, but not too obscure either. I have a friend named Захар.