Praise be to Allah.
Firstly: What is the reward of fasting 'Ashura?
Fasting the day of 'Ashura does expiate for the past year, because the Prophet (peace and blessings of Allah be upon him) said: “Fasting the day of ‘Arafah I hope Allah will expiate thereby for the year before it and the year after it, and fasting the day of 'Ashura I hope Allah will expiate thereby for the year that came before it.” Narrated by Muslim, 1162. This is by the bounty that Allah bestows upon us, whereby fasting one day expiates for the sins of a whole year. And Allah is the Owner of great bounty.
The Prophet (peace and blessings of Allah be upon him) used to be very keen to make sure he fasted on the day of 'Ashura because of its great status. It was narrated that Ibn ‘Abbaas (may Allah be pleased with him) said: I never saw the Prophet (peace and blessings of Allah be upon him) so keen to make sure he fasted any day and preferring it over another except this day, the day of 'Ashura, and this month – meaning Ramadaan. Narrated by al-Bukhaari, 1867.
What is meant by being keen to make sure he fasted it is so as to earn its reward.
Secondly: Why did the Prophet fast the day of 'Ashura?
With regard to the reason why the Prophet (peace and blessings of Allah be upon him) fasted on the day of 'Ashura and urged the people to do likewise is mentioned in the hadeeth narrated by al-Bukhaari (1865) from Ibn ‘Abbaas (may Allah be pleased with him), who said: The Prophet (peace and blessings of Allah be upon him) came to Madeenah and saw the Jews fasting on the day of 'Ashura. 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 said, “We are closer to Moosa than you.” So he fasted on this day and told the people to fast.
The words “this is a good day” – according to a version narrated by Muslim, “This is a great day when Allah saved Moosa and his people and drowned Pharaoh and his people.”
The words “so Moosa fasted on this day” – Muslim added in his report: “In gratitude to Allah, so that is we fast on this day.”
According to another version narrated by al-Bukhaari, “So we fast it out of respect for it.”
The words “and told the people to fast” – according to another version narrated by al-Bukhaari, “He said to his companions, ‘You are closer to Moosa than them, so fast this day.”
Thirdly: What are the sins that are expiated by fasting 'Ashura?
The expiation of sins that is achieved by fasting 'Ashura refers to minor sins; with regard to major sins, they need separate repentance.
Al-Nawawi (may Allah have mercy on him) said:
Fasting the day of ‘Arafaah expiates for all minor sins, in othr words this brings forgiveness for all sins except for major sins.
Then he said:
Fasting the day of ‘Arafaah is an expiation for two years, and the day of ‘Ashoora is an expiation for one year, and if a person’s Ameen coincides with the Ameen of the angels, his previous sins will be forgiven… Each of the things mentioned may bring expiation. If he does something that expiates for minor sins he will be expiated, and if there are no minor or major sins, it will be recorded for him as good deeds and he will rise in status thereby… If there is one or more major sins and no minor sins, we hope that it will reduce his major sins. Al-Majmoo’ Sharh al-Muhadhdhab, part 6.
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said: The expiation of purification, prayer, and fasting Ramadaan, ‘Arafah and 'Ashura applies to minor sins only. Al-Fataawa al-Kubra, part 5.
And Allah knows best.