Source To Learn Halachos Thirty Days Before Rosh Hashanah Gemara Pesachim (6A) tells us that 30 days before Pesach, we begin learning the halachos of Pesach. Ben Yehoyada says the obligation applies to Rosh Hashanah too. Sources: Ben Yehoyada: Eruvin, end of 54b
The month of Elul Abbreviations The month of Elul is known for the many acronyms associated with its name, some of which date back as early as Avudraham and Baal HaTurim. See Mateh Moshe for three acronyms, one from each book of Tanach. Sources: Avudraham: Rosh Hashanah prayers. Baal HaTurim: Devarim 30:6. Mateh Moshe: 5:778.
Custom to Recite L'Dovid in ElulThe Mateh Efraim brings the custom to recite chapter 27 of Psalms from Rosh Chodesh Elul through Shemini Atzeres (Gra didn't say it). In Nusach Ashkenaz, this chapter is recited after Shacharis and Maariv, while in Nusach Sefard, it is recited after Shacharis and Mincha. Various reasons are given for this practice. The Elef L'Mateh suggests that Psalm 27 alludes to Rosh Hashanah, Yom Kippur, and Sukkos. Sources: Mateh Efraim: 581:6. Gra: Masseh Rav 53. Elef L'Mateh: 581:6 based on Midrash Vayikra 21:4.
Why We Say Selichos During ElulThe reason we say Selichos during Elul is because it's an auspicious time for forgiveness. The Tanna Devei Eliyahu says, "From when Moshe ascended to receive the second luchos on Rosh Chodesh Elul until he descended on Yom Kippur, the people fasted and were forgiven." This established Elul as a time of mercy each year.Sources: Tanna Devei Eliyahu: Zuta 4.
When to start SelichosThe Sephardic custom is to recite Selichos throughout the entire month of Elul, beginning from the day after Rosh Chodesh Elul. In contrast, the Ashkenazic custom is to start reciting Selichos from the week of Rosh Hashanah, which this year begins on September 28th. Sources: Shulchan Aruch 581:1.
Ashkenazim only start Selichos closer to Rosh Hashanah While the entire month of Elul is an auspicious time for seeking forgiveness, as we have noted, Ashkenazim begin reciting Selichos only "when it gets close to the Day of Judgment—during the week of Rosh Hashanah" (Levush).This is only true if Rosh Hashanah falls on Thursday or Shabbos; if it falls on Monday or Tuesday, Selichos start a week earlier. Sources: Levush 581:1
Why Start The Previous Week if it falls out on Monday or Tuesday? There are two reasons why we start Selichos the week before rather than the week of Rosh Hashanah when Rosh Hashanah falls on a Monday or Tuesday. The Bais Yosef cites the Mordechai, who says that one must fast (and consequently recite Selichos) for four days before Rosh Hashanah to compensate for the four days of fasting that are not possible during the Ten Days of Repentance. The Levush says that when Rosh Hashanah falls on Monday or Tuesday, we don't have four days during that week to say Selichos. Therefore, we begin saying Selichos the previous week, and in order to avoid confusion, we always start on a Sunday—sources: Bais Yosef (581:2). Levush (581:1) brought in the Taz 581:2.
The second reason why we started Selichos the previous week is that The Eliyahu Rabbah explains why we started Selichos the week before rather than the week of Rosh Hashanah when Rosh Hashanah falls on a Monday or Tuesday. On Rosh Hashanah, we need to make ourselves just like a sacrifice. Each sacrifice needs a minimum of four days of examination to make sure there are no imperfections. Therefore, to be ready for Rosh Hashanah, we need a minimum of four days of Selichos. Sources: Eliyahu Rabbah (581:8), refer to Shemos; 12:3.
The time of Selichos. The ideal times to say Selichos is before sunrise or after midnight (Shulchan Aruch, Igros Moshe). The custom is to say Selichos in the morning, too (Aruch HaShulchan). However, many disapprove of reciting Selichos at night before midnight (Magen Avraham, Sharei Teshuvah, Eliyahu Rabbah). Sources: Shulchan Aruch: 581:1. Igros Moshe: O.C. 2:105. Aruch HaShulchan 581:4. Magen Avraham 565:5. Sharei Teshuvah 581:1. Eliyahu Rabbah 581:9.