The 10th day of the 7th month (Ethanim, or Tishri), the most solemn day of the year. On it all were not only to refrain from work but also to afflict their souls (Lev 23:27-32). This probably included fasting, since in NT times it is evidently this day that is referred to as "the fast" (Acts 27:9). On this day all the sins of the preceding year were finally disposed of in the ceremony of cleansing the sanctuary (16). All who did not afflict their souls on that day were cut off from Israel (ch 23:29). The Day of Atonement was to the Jews a day of judgement. As their tradition later describes it, all are judged on New Year's Day, but those who are not outstandingly good or hopelessly wicked have 9 days more, until the Day of Atonement, before their doom is finally sealed (Talmud Rosh Hashanah 16a).
Another important event connected with the Day of Atonement was the blowing of the trumpet on that day to announce the 50th year of the sabbatical year cycle, the year of jubilee (Lev 25:9, 10). Presumably, then, the sabbatical years, running in the same series with the jubilee years, also began at that time. The Day of Atonement services represented cleansing from sin and reconciliation to God (ch 16:16, 33, 34). The ritual began with the high priest bathing his body and putting on the holy linen garments (v 4). For himself and his house he offered a bull for a sin offering (v 6). After this personal preparation a goat designated "for the Lord," previously chosen by lot from two acquired for the service (vs. 5, 7, 8), was sacrificed (v 9). Then, amid clouds of incense ascending from the altar before the second veil (vs. 12, 13) the high priest entered the Most Holy Place and sprinkled the blood first of the bull, then of the goat, upon and before the mercy seat (v 15), which covered the ark containing, among other things, the tables of the Decalogue (Heb 9:4). In this manner the holy place was cleansed, and atonement made for the sins of the people (Lev 16:16). In a similar manner the altar was cleansed (vs. 18, 19). Later, but not until the work of reconciliation for the holy place, the altar, and the people was ended (v 20), the transgressions of the people were transferred ritually to the goat, described as being "for Azazel," (v 10, RSV). This goat was then led into the wilderness (vs. 20-22).
The high priest was a type of Christ, the high priest in the heavenly sanctuary (Heb 8:1). The earthly priest performed his services "unto the example and shadow of heavenly things" (v 5). The author of Hebrews explains that by the high priest's entering only once a year into the second apartment the Holy Spirit signified that "the way into the holiest of all was not yet made manifest, while as the first tabernacle was yet standing" (ch 9:8) -- Seventh-day Adventist Bible Dictionary.