From Rosh Chodesh Elul until after Yom Kippur are days of Divine favor. Even though throughout the entire year the Holy One, blessed is He, accepts the repentance of those who return to Him wholeheartedly, nevertheless, these days are unexcelled and most suitable for repentance, because they are days of mercy and favor. On Rosh Chodesh Elul, Moshe went up Mount Sinai to receive the Second Tablets; he remained there for forty days, and came down on the tenth day of Tishrei when the atonement was completed. From then on these days have been designated as days of Divine favor and the tenth day of Tishrei as Yom Kippur.....The interpreters of allusions also commented: It is written (Deuteronomy 30:6) "And Hashem your God will circumcise your heart and the heart of your children;" the initials of the words [es levavecho ve'es levav] form the acronym Elul. Also, the initials of Ani ledodi vedodi li, ["I am my Beloved's and my Beloved is mine," form the acronym Elul. Also, the initials of Ish lerei'eihu umatanos la'evyonim ["One to another and gifts to the poor"] (Esther 9:22) form the acronym Elul. These acronyms are an allusion to three things: Repentance, Prayer and Charity which must be practiced zealously during this month. "Hashem will circumcise etc." alludes to repentance, "I am my Beloved's etc." alludes to prayer, for prayer is the song of love. "One to another and gifts to the poor," alludes to charity.

(1) We have the custom to get up at the break of dawn to recite Selichot and tachanunim from Rosh Chodesh Elul until Yom Kippur. (Rema: But the custom of the Ashkenazi Jews is not so. Rather, from Rosh Chodesh on, we blow the Shofar after Shacharit, and there are some places where they also blow the Shofar after Arvit. We get up at dawn to recite Selichot from the Sunday before Rosh HaShanah on; and if Rosh HaShanah falls on a Monday or Tuesday, we begin reciting Selichot from Sunday the week before Rosh HaShanah....

Rabbi Joshua, son of Ḳorchah, said: Forty days was Moses on the mountain, reading the Written Law by day, and studying the Oral Law by night. After the forty days he took the tables (of the Law) and descended into the camp on the 17th of Tammuz, and he broke in pieces the tables, and slew the sinners in Israel. He then spent forty days in the camp, until he had burnt the calf, and powdered it like the dust of the earth, and he had destroyed the idol worship from Israel, and he instituted every tribe in its place. And on the New Moon of Elul the Holy One, blessed be He, said to him: "Come up || to me on the mount" (Ex. xxiv. 12), and let them sound the Shofar (ram's horn) throughout the camp, for, behold, Moses has ascended the mount, so that they do not go astray again after the worship of idols. The Holy One, blessed be He, was exalted with that Shofar, as it is said, "God is exalted with a blast, the Lord with the sound of the shofar" (Ps. xlvii. 5). Therefore the sages instituted that the Shofar should be sounded on the New Moon of Ellul every year.

...One who wishes to merit repentance, should accustom himself in saying Psalms, for the recitation of Psalms is effective (mesuggal) for repentance. ... By reciting Psalms, even one who has no independent awakening to repentance, will be aroused to repent, and will also merit by means of Psalms to reach the gate and the letter that are appropriate for him and to open the gate. Thus, by means of Psalms he will merit to repent.

This is conceptually linked ("bechinat") to (II Samuel 23:1) "The utterance of the man set on high... (huqam `al)" (concerning which) our Rabbis expounded (bMoed Katan 16b), that he [David] raised up the yoke (heqim `ullah) of repentance, and [the same verse continues,] "...the favorite of the songs of Israel," for it is a result of the aspect of...the book of Psalms that he wrote, that one merits repentance, as said above. The Rabbis stated (bAvodah Zarah 4b) that David was not fit to act as he did in that matter (the affair of Batsheva, 2 Samuel 11), except that in this way he could teach repentance to the individual (sinner). Consequently, the central teaching of repentance is through King David, and the root of the repentance of King David is the Book of Psalms, which he said in a state of very great awakening and with the Holy Spirit (Ruach haKodesh), so that everyone, according to his capacity, can find himself (his own spiritual situation) in the Book of Psalms, in order to merit repentance through the recitation of Psalms, as stated above.