Elul is an auspicious month for examining one’s deeds and making Teshuva. The letters of the word Elul in Hebrew correspond to the initials of the verse “Ani LeDodi VeDodi Li” (Shir HaShirim 6:3), which means that Hashem is especially close to us during this month. The numerical value of the Sofe Tevot (the last letter of each word) of this verse is 40, corresponding to the number of days that Moshe Rabbenu went up to Mount Sinai to beseech Hashem for atonement. This is why we recite Selihot for 40 days-commencing in the beginning of Elul and culminating on Yom Kippur-just as Moshe Rabbenu did when receiving the second Tablets.
The Arizal writes (Pri Etz Haim, Shaar Rosh Hashana, ch. 1) that the letters of Elul correspond to the initials of the verse “[VaAsher Lo Tzada VeHaElokim] Ina Leyado Vesamti Lecha [Makom Asher Yanus Shama]” (Shemot 21:13). In context, this verse explains that cities of refuge are to be provided for one who inadvertently killed one’s fellow as a form of protection from vengeful relatives. Therefore, Elul is symbolically a month in which we can seek refuge from any sins we may have committed inadvertently, and to ask for forgiveness for them. The Arizal also writes that Elul also stands for “Ye’alelun Et Ar’a”, the Aramaic translation of the verse dealing with the Spies’ scouting of the Land of Israel. As such, Elul is a month in which we need to “scout out” and take stock of our positive and negative deeds from the previous year. Finally, Sefer Amarcal explains that Elul stands for “Ish Le’Re’ehu Umatanot LaEvyonim”, the verse from Megilat Esther that talks about the gift-giving that happens on Purim, which also alludes to the fact that in Elul one should increase one’s good deeds to one’s fellows.
Summary: Elul is a time when one should increase one’s good deeds and examines one’s previous actions, in preparation for Rosh Hashana.