The Day of Ashura (Arabic: عاشوراء ʻĀshūrā’ , colloquially: /ʕa(ː)ˈʃuːra/; Urdu: عاشورہ; Persian: عاشورا; Turkish: Aşure Günü) is on the tenth day of Muharram in the Islamic calendar and marks the climax of the Mourning of Muharram.
It is commemorated by Shi'a Muslims as a day of mourning for the martyrdom of Husayn ibn Ali, the grandson of Muhammad at the Battle of Karbala on 10 Muharram in the year 61 AH (in AHc: October 9, and in AHt: October 10, 680 CE). In some Shi'a regions of Muslim countries such as Afghanistan, Iran, Turkey, Azerbaijan, Iraq, Lebanon, Bahrain, and Pakistan, the Commemoration of Husayn ibn Ali has become a national holiday and most ethnic and religious communities participate in it. Even in predominantly Hindu majority but secular country like India, Ashura (10th day in the month of Muharram) is a public holiday.
Shia commonly believe that taking part in Ashura is to be absolved of sin. A popular Shia saying has it that, `a single tear shed for Husayn washes away a hundred sins.`
It is commemorated by Sunni Muslims (who also refer to it as The Day of Atonement) as the day on which the Israelites were freed from the Pharaoh (called 'Feron' in Arabic) of Egypt. According to Sunni Muslim tradition, Ibn Abbas narrates that Muhammad came to Madina and saw the Jews fasting on the tenth day of Muharram. He asked, “What is this?” They said, “This is a good day, this is the day when Allah saved the Children of Israel from their enemy and Musa (Moses) fasted on this day.” So he fasted on this day and told the people to fast. Many Sunnis also recognize the importance of the events at Karbala and the martyrdom of Imam Husayn in regards to Ashura.
See the following links from GotQuestions.org:
What does it mean that Jesus died for our sins?
Why did the sacrificial system require a blood sacrifice?
What is the Passover Lamb? How is Jesus our Passover Lamb?
What did Jesus mean when He said “It is finished”?