The month of Elul, which began Tuesday evening, is the final one of the Jewish year. It is immediately followed by Rosh HaShana, the beginning of the new year and a Day of Judgment.
For this reason it is considered a time of soul-searching, in which the Israelites are expected to undergo a process of 'teshuva' – a return to roots and to morality, and a process of renewal in anticipation of the new year.
The 40 days that include Elul and the ten days that follow it are considered “Days of Mercy” because they are the days in which Moses – to whom Jews usually refer as Moshe Rabbenu, or “Our Rabbi Moses” – ascended Mount Sinai and prayed for the Israelites after the Sin of the Calf. On Yom Kippur, G-d told him that He had forgiven Israel's transgression and Moses went back down the mountain with the new Tablets of commandments.
The days are therefore considered fit for teshuva and tikkun – “fixing” one's heart and mending one's ways.
Sephardic Jews traditionally recite the selichot, or prayers of penitence, throughout the month, either in the very early morning or after midnight. The selichot are prayers to G-d asking that He will forgive us our trespasses, bring us closer to Him, have mercy on Nation of Israel and fully redeem them.
Ashkenazi Jews begin to recite selichot on the week that precedes Rosh Hashana. Some Yemenite Jews begin to recite them in mid-Elul.
Ashkenazi Jews – descendants of communities that lived in Europe for hundreds of years – also add Psalm 2, which begins with the words “A Psalm of David: G-d is my light and my salvation; whom shall I fear?” – following their prayers during Elul. It is also customary to blow the shofar, or ritual ram's horn, after the morning prayers, to arouse the hearts of the believers to make teshuva before Judgment Day.