In His loving kindness, the Holy One, blessed be He, does not turn away from us when we break one of His laws.  Instead, He orders us to return to Him at any time we stray from the path of Torah and mitzvot.  G-d is always ready and willing to accept our sincere repentance (teshuva), but even more so during the month of Elul.

In fact, the forty days, from Rosh Chodesh Elul until Yom Kippur, the Day of Atonement, have been set aside as a period of special grace and favour; it is the best time to look into our way of life and our conduct during the year about to end, and make up our mind to return more closely to G-d.  During this time, G-d is especially gracious to us, and willing to forgive us for any wrongdoing in the past.

This period of special grace and favour goes back to the time when we first became a people, on receiving the Torah at Mount Sinai.  When the children of Israel worshiped the Golden Calf, and Moses broke the Tablets with the Ten Commandments (17th of Tammuz), the children of Israel had committed the greatest possible sin.  Yet, during the forty days from Rosh Chodesh Elul until Yom Kippur, G-d forgave them.  It was on Rosh Chodesh Elul that Moses went up again on Mount Sinai, and it was Yom Kippur when he came down, with the second Tablets and with the message of G-d's forgiveness.  From then on these forty days became the time of G-d's grace and favour, more so than any other time in the year.

Sounding of the Shofar

When Moses went up on Mount Sinai on Rosh Chodesh Elul, the shofar was sounded every day until he came down again; in order to remind the people that Moses was on Mount Sinai praying for them, and that they, too, should pray for G-d's forgiveness.  It is therefore customary to sound the shofar during all of the month of Elul, except on Shabbos, and except also on the last day of the month -- erev Rosh Hashanah.  On Shabbat the shofar is never sounded (not even on Rosh Hashanah), and on the last day of Elul the shofar is not sounded in order to make an interval between the sounding of the shofar during the month of Elul which is merely custom, and that of Rosh Hashanah which is a commandment in the Torah.