On the early morning of the 1st of Elul of the year 2448 from creation (1313 BCE) Moses ascended Mount Sinai, taking with him the stone tablets he had hewn by divine command (see "Today in Jewish History" for yesterday, Av 30), 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).
Moses remained on the mountain for 40 days, 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 following their betrayal of the covenant between them with their worship of the Golden Calf. This was the third of Moses' three 40-day periods on Mount Sinai in connection with the Giving of the Torah. Ever since, the month of Elul serves as the "month of Divine mercy and forgiveness."
Font: chabad.org