As the last month of the Jewish year, Elul is traditionally a time of introspection and stocktaking -- a time to review one's deeds and spiritual progress over the past year and prepare for the upcoming "Days of Awe" of Rosh HaShanah and Yom Kippur.
As the month of Divine Mercy and Forgiveness (see "Today in Jewish History" for Elul 1) it is a most opportune time for teshuvah ("return" to G-d), prayer, charity, and increased Ahavat Yisrael (love for a fellow Jew) in the quest for self-improvement and coming closer to G-d. Chassidic master Rabbi Schneur Zalman of Liadi likens the month of Elul to a time when "the king is in the field" and, in contrast to when he is in the royal palace, "everyone who so desires is permitted to meet him, and he receives them all with a cheerful countenance and shows a smiling face to them all."
Specific Elul customs include the daily sounding of the shofar (ram's horn) as a call to repentance. The Baal Shem Tov instituted the custom of reciting three additional chapters of Psalms each day, from the 1st of Elul until Yom Kippur (on Yom Kippur the remaining 36 chapters are recited, thereby completing the entire book of Psalms). Click below to view today's Psalms.
Chapter 58
Chapter 59
Chapter 60
Elul is also the time to have one's tefillin and mezuzot checked by an accredited scribe to ensure that they are in good condition and fit for use.
Links: More on Elul