In Geonic times, the custom in both of the prominent yeshivot in Babylonia was to recite Seliḥot during the Ten Days of Repentance. In a few places, Seliḥot were recited during the entire month of Elul. By the end of the medieval era, Sephardic communities had accepted the practice of reciting Seliḥot throughout Elul (SA 581:1). On Rosh Ḥodesh Elul itself, though, Seliḥot are not recited (Responsa Rama Mi-Fano §79; Kaf Ha-ḥayim 581:1). As Rosh Ha-shana draws near, more and more people make sure to attend Seliḥot, and people are especially meticulous about doing so during the Ten Days of Repentance. The Ashkenazic custom is to begin reciting Seliḥot on the Saturday night before Rosh Ha-shana, provided that there will be at least four days of Seliḥot prior to the holiday. This means that if Rosh Ha-shana starts on Thursday or Shabbat, Seliḥot begin the Saturday night before the holiday, but if Rosh Ha-shana starts on Monday or Tuesday, Seliḥot begin the previous Saturday night.
While the Sages did not make reciting Seliḥot mandatory, it is the predominant Jewish custom. Nevertheless, one who finds it difficult to wake up for Seliḥot need not do so during Elul. During the Ten Days of Repentance, though, he should make serious efforts to recite Seliḥot, as these days are auspicious for repentance and forgiveness. (See Rosh Ha-shana 18a; MT, Laws of Repentance 2:6.) If one is unable to go to sleep early, and waking up for Seliḥot would result in his being too exhausted to fulfill his work obligations, then he should not wake up early, even during the Ten Days of Repentance. Rather, he should try to recite chapters of Tehilim, and during the course of the day he can recite those sections of the Seliḥot that an individual may recite alone. (See section 7 below.)
The accepted ruling is that even very diligent Torah scholars should recite Seliḥot (Birkei Yosef and Sha’arei Teshuva 581:1). Indeed, this is the custom in all yeshivot, even though reciting Seliḥot takes time away from Torah study. However, if one finds that waking up early makes him lose even more time than the time taken by the recitation of Seliḥot because he cannot concentrate on his studies later on, it is better for him not to wake up early for Seliḥot.
Why then do we always begin Seliḥot on Saturday night rather than beginning four days before Rosh Ha-shana? Some suggest that always beginning Seliḥot on the same night is less confusing (Tur and Rema 581:1; MB ad loc. 6). Others suggest that the point of beginning Seliḥot on Saturday night is in order to start our supplications as we exit the holy Shabbat. During Shabbat, people study Torah joyfully, which allows the Shekhina – which only rests where there is the joy of a mitzva – to rest upon them (Leket Yosher).