The month of Elul is a time of repentance in preparation for the High Holy Days of Rosh Hashanah and Yom Kippur. In Aramaic (the language spoken by Jews living at the time that the months were given names), the word “Elul” means “search.” The Talmud writes that the Hebrew word "Elul" can be expanded as an acronym for "Ani L'dodi V'dodi Li" - "I am to my Beloved and my Beloved is to me." Elul is seen as a time to search one's heart and draw close to God in preparation for the coming Day of Judgement, Rosh Hashanah, and Day of Atonement, Yom Kippur.
During the month of Elul, there are a number of special rituals leading up to the High Holy Days. It is customary to blow the shofar every morning (except on Shabbat) from Rosh Hodesh Elul (the first day of the month) until the day before Rosh Hashanah. The blasts are meant to awaken one's spirits and inspire him to begin the soul searching which will prepare him for the High Holy Days. As part of this preparation, Elul is the time to begin the sometimes-difficult process of granting and asking for forgiveness. It is also customary to recite Psalm 27 every day from Rosh Hodesh Elul through Hoshanah Rabbah on Sukkot (in Tishrei).
Aside from the blowing of the shofar, the other major ritual practice during Elul is to recite selichot (special penitential prayers) either every morning before sunrise during the week before the last Wednesday before Rosh Hashanah (Ashkenazi tradition) or every morning during the entire month of Elul (Sephardi tradition). Ashkenazi Jews begin the recitation of selichot with a special service between solar mid-night (not 12:00) and morning light on the first day of Selichot.
Many Jews also visit the graves of loved ones throughout the month in order to remember and honor those people in our past who inspire us to live more fully in the future.
Another social custom is to begin or end all letters written during the month of Elul with wishes that the recipient have a good year. The standard blessing is "K'tiva V'Hatima Tova" ("a good writing and sealing [of judgement]"), meaning that the person should be written and sealed in the Book of Life for a good year. Tradition teaches that on Rosh Hashanah, each person is written down for a good or a poor year, based on their actions in the previous one, and their sincere efforts at atoning for mistakes or harm. On Yom Kippur, that fate is "sealed."