Jewish Prayers: Selichot
Selichot are special prayers for forgiveness, said on fast days and also during the period preceding Yom Kippur.
At the Selichot service, worshipers begin to examine their deeds of the past year, seeking forgiveness from G-d and promising to improve their behavior in the New Year. The prayers are specifically tailored to help worshipers direct their hearts and minds to the process of teshuvah (Hebrew for repentance).
In the Sephardic tradition, Selichot are said from the beginning of the month of Elul, while in the Ashkenazic tradition Selichot are begun from the Sunday (often the Saturday night) before Rosh Hashanah until Yom Kippur. If Rosh Hashanah begins on a Monday or Tuesday, however, selichot begins on the Sunday of the week before Rosh Hashanah, to make sure that there are at least three days of Selichot.
In general, the proper time to say Selichot are at the end of the night, just before the morning, since this time is considered, according to Jewish Mysticism, as especially favorable in terms of the presence and closeness of God. Hence, selichot are typically recited in the early morning, before the daily shacharit service.
The first night of Selichot is different from the other days. First, it is customary to say Selichot the first night before going to sleep, and, since the first part of the night is considered a time of din, judgment, the Selichot are not recited on the first night until after chatzot, relative midnight. A person should consult a Jewish calendar or their rabbi to determine the specific time of chatzot for their area.
A fundamental part of the selichot service is the repeated recitation of the "Thirteen Attributes," a list of G-d's thirteen attributes of mercy that were revealed to Moses by G-d after the sin of the golden calf (Exodus 34:6-7): "Ha-shem , Ha-shem , G-d , merciful , and gracious , long-suffering , abundant in goodness  and truth , keeping mercy unto the thousandth generation , forgiving iniquity  and transgression  and sin , who cleanses ." Why is "Ha-shem" listed twice as an attribute? And why are three of these "attributes" Names of G-d? Different names of G-d connote different characteristics of Him. The four-letter Name of G-d (rendered here as "Ha-shem," literally "the name") is the Name used when G-d is exhibiting characteristics of mercy, and the Talmud explains that this dual usage indicates that G-d is merciful before a person sins, but is also merciful after a person sins. The third attribute is a different Name of G-d that is used when G-d acts in His capacity as the almighty ruler of nature and the universe. G-d appeared to Moses and taught him these Thirteen Attributes saying, "Whenever Israel sins, let them recite this in its proper order and I will forgive them." Thus, this appeal to G-ds mercy reassures that repentance is always possible, and that G-d always awaits a return to Him. The implication is also that if people emulate G-ds merciful ways, He will treat them mercifully in return.
Sources: Judaism 101; The Orthodox Union; The Jewish Appleseed Foundation