Wikipedia writes:
Ashura (Day of Remembrance) is the tenth day of Muharram in the Islamic calendar. ...
What is the significance of the day of Ashura, and is it obligatory to fast that day?
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Ashura (Day of Remembrance) is the tenth day of Muharram in the Islamic calendar. ...
What is the significance of the day of Ashura, and is it obligatory to fast that day?
Based on the Hadith of Prophet Muhammad (saws), Sunni Muslims celebrate Ashura as the day when Prophet Moses (Moosa) fasted on that day because Allah saved the Israelites from their enemy in Egypt. Bukhari narrates:
Ibn ‘Abbaas, who said: The Prophet (saws) came to Madinah and saw the Jews fasting on the day of Ashoora. He said, “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 Moosa fasted on this day.” He (the Prophet Muhammad) said, “We are closer to Moosa (Prophet Moses) than you.”
al-Bukhari (1865)
According to a version narrated by Muslim:
This is a great day when Allah saved Moosa (Moses) and his people and drowned Pharaoh and his people.”
This is why the Prophet told us to fast on the day of Ashura.
He fasted on that day and commanded Muslims to fast on this day. (Al-Bukhari)
The prophet observed the fast on Ashura (the 10th of Muharram), and ordered (Muslims) to fast on that day.
(Agreed upon Hadith i.e. Bukhari & Muslim).
The following year, Allah commanded the Muslims to fast the month of Ramadan, and the fasting of 'Ashura' became optional. It is also reported that the Prophet (peace and blessings be upon him) intended to fast on the ninth and tenth.
Ibn 'Abbas reported: The Messenger of Allah fasted on the day of 'Ashura' and ordered the people to fast on it. The people said, "O Messenger of Allah, it is a day that the Jews and Christians honor." The Prophet said, "When the following year comes, Allah willing, we shall fast on the ninth." The death of the Prophet came before the following year.
(Muslim and Abu Dawud)
Fasting on this day also is a means of expiation of sins.
Narrated by Abi Katada: The prophet was asked about fasting on Ashura (the 10th of Muharram), he said: “it expiates the previous year (for sins).”
(Sahih Muslim)
It was proven from the Prophet (peace and blessings of Allah be upon him) that the best fasting after Ramadan is fasting in the month of Muharram.
It was narrated that Abu Hurayrah (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “The best fasting after Ramadan is the month of Allah, Muharram, and the best prayer after the obligatory prayer is prayer at night.”
Narrated by Muslim, 1163.
It is not obligatory to fast on this day as Allah made Ramadan obligatory to fast.
When the Messenger of Allah (peace and blessings of Allah be upon him) came to Madinah, he told the Muslims to fast on three days of every month and on the day of 'Ashura', then Allah made fasting obligatory when He said (interpretation of the meaning): " … observing the fasting is prescribed for you … " [Al-Qur'an 2:183]
[Ahkam al-Qur'an by al-Jassas, part 1]
Shia Muslims’ observance of Ashura is different altogether. They observe Ashura as the day of martyrdom of Hussein ibn Ali, the grandson of Prophet Muhammad at the Battle of Karbala.
Shia Muslims, therefore, consider this a day of sorrow and observe it as such by refraining from music, listening to sorrowful poetic recitations, wearing mourning attire, and refraining from all joyous events (e.g. weddings) that in anyway distract them from the sorrowful remembrance of that day.
(Read here about Sunni Shia History and Conflicts)
Husayn's grave became a pilgrimage site among Shia Muslims only a few years after his death. A tradition quickly developed of pilgrimage to the Imam Husayn Shrine.
On Ashura, some Shi'as observe mourning with a blood donation, which is called "Qame Zani", and flailing.
Traditional flagellation rituals such as Talwar zani (talwar ka matam or sometimes tatbir) use a sword. Other rituals such as zanjeer zani or zanjeer matam involve the use of a zanjeer (a chain with blades).
These religious customs show solidarity with Husayn and his family. Through them, people mourn Husayn's death and express regret for the fact that they were not present at the battle to fight and save Husayn and his family.
Some Shi'a believe that taking part in Ashura washes away their sins. A popular Shi'a saying has it that "a single tear shed for Husayn washes away a hundred sins".
(Read more here)
And Allah knows best.