Why do we blow Shofar in Elul?
The Tur (O.C. 581) brings down two reasons the Pirkie Drav Eliezer says when Moshe went up on Rosh Chodesh Elul to get the second Luchos in order to prevent the Jews from doing something like the golden calf the Shofar was blown. This based on a Posuk in Tehillim God shall be exalted with the trumpet blast; the Lord, with the sound of the Shofar (47:6). Then he brings a Posuk in Amos (3:6) Will a Shofar be sounded in the city and the people not quake. This is because the first Posuk indicates a prevention of further Aveiros the second Posuk symbolizes that they will do Teshuvah. The Shlah in his Chiddushim on Gemarah Rosh Hashanah says the Pirkie Drav Eliezer is only a reason to Blow on Rosh Chodesh itself the rest of the month is because of the Posuk in Amos. The Levush (581) adds his own twist saying it is the concept of Turn away from evil and do good on Rosh Chodesh it is turn away from evil then on it is do good.The significance of the Month time of blowing's source is from Tiku baChodesh shofar since Chodesh is a month.