Seven weeks after the exodus, the people of Israel received the Torah at Mount Sinai and entered into an eternal covenant with G‑d as His chosen people. But just 40 days later, while Moses was still up on the mountain, they violated their special relationship with G‑d by worshipping a golden calf. Upon descending from the mountain and witnessing their transgression, Moses smashed the two stone tablets on which G‑d had inscribed the Ten Commandments; he then retuned to Mount Sinai for a second 40 days to plead with G‑d on Israel's behalf.
On the early morning of the 1st of Elul, Moses once again ascended Mount Sinai, taking with him the stone tablets he had hewn, by divine command, for G‑d to re-inscribe the Ten Commandments. On the mountain, G‑d allowed Moses to "see My back, but not My face" (which Maimonides interprets as a perception of G‑d's reality but not His essence) — the closest any human being ever came to knowing G‑d — and taught him the secret of His "Thirteen Attributes of Mercy" (Exodus 33:18-34:8).
For the third time, Moses remained on the mountain for 40 days, from the 1st of Elul until the 10th of Tishrei (Yom Kippur), during which time He obtained G‑d's whole-hearted forgiveness and reconciliation with the people of Israel. Ever since, the month of Elul serves as the "month of Divine mercy and forgiveness."
Source:
https://www.chabad.org/holidays/JewishNewYear/template_cdo/aid/4351/jewish/A-Brief-History-of-Elul.htm
They are four days in the year that serve as the New Year, each for a different purpose: On the first of Nisan is the New Year for kings; it is from this date that the years of a king’s rule are counted. And the first of Nisan is also the New Year for the order of the Festivals, as it determines which is considered the first Festival of the year and which the last. On the first of Elul is the New Year for animal tithes; all the animals born prior to that date belong to the previous tithe year and are tithed as a single unit, whereas those born after that date belong to the next tithe year. Rabbi Elazar and Rabbi Shimon say: The New Year for animal tithes is on the first of Tishrei.
[The days] from Rosh Chodesh Elul until after Yom Kippur, are days of Divine favor [and acceptance.] 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 acceptance,] and the tenth day of Tishrei as Yom Kippur [Day of Atonement]. In most communities it is the custom to fast on the day before Rosh Chodesh Elul and to recite the prayers of Yom Kippur Katan [minor Yom Kippur], in order to be spiritually prepared for repentance. If Rosh Chodesh occurs on Shabbos, Yom Kippur Katan is held on the preceding Thursday.
The Ari (Rabbi Yitzchak Luria), of blessed memory, wrote, "If he did not lie in ambush but Hashem made it happen, then I will provide …" (Exodus 21:13) The initials of the words [ina le'yado vesamti lecha] form the acronym Elul, to indicate that this month is a favorable time for repentance to be accepted for the sins committed during the entire year. It also alludes to the fact that sins done inadvertently also require repentance during this month. 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,"] [Song of Songs 6:3] 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.
https://www.washingtonpost.com/lifestyle/magazine/pearls-before-breakfast-can-one-of-the-nations-great-musicians-cut-through-the-fog-of-a-dc-rush-hour-lets-find-out/2014/09/23/8a6d46da-4331-11e4-b47c-f5889e061e5f_story.html
The LORD is my light and my help;
whom should I fear?
The LORD is the stronghold of my life,
whom should I dread? (2) When evil men assail me
aOr “to slander me”; cf. Dan. 3.8; 6.25.to devour my flesh-a—
it is they, my foes and my enemies,
who stumble and fall. (3) Should an army besiege me,
my heart would have no fear;
should war beset me,
still would I be confident.
(4) One thing I ask of the LORD,
only that do I seek:
to live in the house of the LORD
all the days of my life,
to gaze upon the beauty of the LORD,
bMeaning of Heb. uncertain.to frequent-b His temple. (5) He will shelter me in His pavilion
on an evil day,
grant me the protection of His tent,
raise me high upon a rock. (6) Now is my head high
over my enemies round about;
I sacrifice in His tent with shouts of joy,
singing and chanting a hymn to the LORD.
(7) Hear, O LORD, when I cry aloud;
have mercy on me, answer me. (8) bMeaning of Heb. uncertain.In Your behalf-b my heart says:
“Seek My face!”
O LORD, I seek Your face. (9) Do not hide Your face from me;
do not thrust aside Your servant in anger;
You have ever been my help.
Do not forsake me, do not abandon me,
O God, my deliverer. (10) Though my father and mother abandon me,
the LORD will take me in. (11) Show me Your way, O LORD,
and lead me on a level path
because of my watchful foes. (12) Do not subject me to the will of my foes,
for false witnesses and unjust accusers
have appeared against me. (13) Had I not the assurance
that I would enjoy the goodness of the LORD
in the land of the living…
(14) Look to the LORD;
be strong and of good courage!
O look to the LORD!