The days from ראש חדש אלול until after יום כיפור are called ימי רצון; the days of redemption. These days are specifically designated for doing תשובה. You might think that everyday should be תשובה day. So, what's so special about ימי רצון? The answer is that אלול is designated for doing תשובה because that is when 'ה forgave בני ישראל for חטא העגל.
Rosh Chodesh Elul has special significance in regard to “second chances.” On Rosh Chodesh Elul, Moshe went back up Har Sinai to receive the second set of Luchos. The second set replaced the first set of Luchos that was broken because of the collective sins of Klal Yisrael. Moshe Rabbeinu spent 40 days and 40 nights there, returning on the tenth day of Tishrei — Yom Kippur. Since that time, these days have been considered yemei ratzon — days in which a person’s teshuvah is more readily accepted.
We can better prepare ourselves for judgment during this time by fulfilling the mitzvos of teshuvah, tefillah and tzedakah.
Each morning during the month of Elul, the shofar is sounded after Shacharis. According to the Gemara in Rosh Hashanah, Moshe Rabbeinu enacted that the shofar be sounded during this time so that Klal Yisrael will not sin again during this time. This minhag helps us in our teshuvah. Rav Moshe Feinstein, zt”l, rules (Igros Moshe OC IV #21) that if a shul forgot to blow shofar at Shacharis they should do so during Minchah. By Rabbi Yair Hoffman for the Five Towns Jewish Times http://www.theyeshivaworld.com/news/headlines-breaking-stories/337693/elul-an-overview.html#sthash.VxdePVM0.dpuf