This is not a feast, but a fast. It has become well known as Yom Kippur (Day of Atonement), because Israel was attacked on Yom Kippur (October 6, 1973 which was the 10th of Tishri of the Hebrew year 5734) in what became known as the Yom Kippur war.
The days between the Feast of Blowing and the Day of Atonement (both inclusive) are known as the Days of Awe. The Jews believe that this time has been set aside by God for them to collectively repent as a nation, for personal as well as national sins.
Kippur means literally covering, and was God’s annual one-year provision for covering of the sins of His people. The blood of the Sacrifice covered the sins, but did not remove them.
It should be noted that there were two sins which Yom Kippur did not cover:
The first was idolatry: When a sinner turned from the One who provided the covering to worship a false god, it resulted in the sinner being denied the covering.
The second was the sin of rebellion (committed, for instance, by King Saul). There was no possibility of atonement or covering for the sin of rebellion. For the Lord said, “Obedience is better than sacrifice.”
We believe that after the Rapture, the Jews will be conquered and ruled by the Beast for 3 and a half years, but on the Day of Atonement of that year a collective realization will come upon them that they had after called caused the Messiah Jesus to be put to death by their rejection of Him. They will then mourn for Him whom they have pierced, fulfilling the prophecy of Zechariah 13.