1) The custom is to blow the Shofar during the month of Elul. The Ashkenazic custom is to blow it after the morning services. Amongst the Sefardim, some blow when reciting the thirteen attributes of hashem, others at the Kaddish before “te’anu vetit’aru”, and others do not blow at all. Whatever custom one has, on the eve of the New Year one does not blow, in order to distinguish between the voluntary and obligatory blowing.
2) The reason for the blowing is to arouse the heart of the people to repent, for the nature of the Shofar is to arouse and terrorize, as the verse says “If a Shofar will be blown in a town and the people will not be afraid”. (Kitzur Shulchan Aruch 128)