From Rosh Hodesh Elul until after Yom Kippur is a time of (Divine) favor. Although, throughout the entire year, the Holy One, blessed be He, accepts the repentance of those who return to Him wholeheartedly, in any case, these days (between the 1st of Elul and Yom Kippur) are more special have been set aside for repentance, because they are days of mercy and favor. Because on Rosh Hodesh Elul, Moses ascended Mount Sinai to receive the second set of stone tablets and spent forty days there, descending on the tenth of Tishrei, which was then the finish of the atonement period1 From then on, these days were set aside as days of favor, and the tenth of Tishrei as the Day of Atonement. 2 It is the custom in most places to fast on the day preceding Rosh Hodesh Elul and follow the order of prayer (termed) the Minor Day of Atonement to prepare one's heart to repent. If Rosh Hodesh Elul falls on Shabbat, one advances (the fast) to the Thursday before it. The Ari (lit; The Rabbi, our master, Rabbi Isaac Luria), may his memory be a blessing,3 wrote:4 ''Surely (a man) does not lie in ambush (to murder), but G-d (who) caused this to happen, then I will provide for you (a place for you to take refuge)'' the first letters (of the words spell) Elul. To tell (us) that this month is (like a place of refuge,) a time of favor when (G-d) accepts (our) repentance on the sins done throughout the year. It also alludes to (sins) committed inadvertently that also need repentance during this month The Rabbis5 have also noted6 ''And G-d, your Lord, will circumcise your heart and the hearts of your descendants'' the first letters (of the words spell) Elul and also7 ''I am for my Beloved and my Beloved is for me'' the first letters (of the words spell) Elul and also8 ''a person (sending gifts) to his friend and (giving) presents to the poor.'' the first letters (of the words spell) Elul (The verses) allude to the three things, which are repentance prayer, and charity, which must be done eagerly in this month. ''And G-d will circumcise...'' alludes to repentance; ''I am for my Beloved...'' alludes to prayer, which is ''a song of lovers''; ''Sending gifts...'' alludes to charity. (KSA 128:1)
1) For the sin of the Golden Calf.
2) Moses went up to Mt Sinai for 3 consecutive 40 day periods; for the first forty days, Hashem taught him the entire Torah and gave him the first set of stone tablets, which Moses subsequently smashed when he came down the mountain and saw the Jews dancing around the Golden Calf; Moses then spent the next 40 days begging Hashem to forgive the Jewish People; the final forty days was spent receiving the second set of tablets.
3) Rabbi Isaac Luria (1534-1572), renowned as the greatest kabbalist of modern times, is commonly known as the ARI, an acronym standing for Elohi Rabbi Yitzhak - the G-dly Rabbi Isaac.
4) To explain the verse Exodus 21:13.
5) lit; the explainers of lists.
6) Deuteronomy 30:6
7) Song of Songs 6:3
8) Esther 9:22