Ashura--The 10th of Muharram--commemorates a lot of important historical instances in Islam, few of which are as follows -
Martyrdom of Hussain Ibn Ali (RA)
Birth of Prophet Ibrahim (AS)
Repentance of Prophet Adam (AS), Prophet Dawud (AS), and the people of Prophet Yousuf (AS) was accepted
Prophet Nuh (AS)'s arc landed on mount Judi and they had kept fast
River Nile was split by the command of ALLAH SWT for the children of Israel (AS) in order to save them from Pharaoh under the prophethood of Musa (AS)
Prophet Younus (AS) came out of the belly of fish
Prophet Isa (AS) was born and also ascended to the heaven on this day
Prophet Yaqoob (AS)'s eyesight was restored, and so on and so forth
Our Prophet Mu'hammed (P.B.U.H) have instructed us to fast on the day of Ashura (10th Muharram) as this expiates the sins of the past year. Prophet Mu'hammed (PBUH) used to fast on 9th (as well) and 10th Mu'harram in order to distinguish from the Jews (The children of Israel AS). Following are some of the narration translations in support of the aforementioned ahkaam -
Sahih Bukhari, Volume 3, Book 31, Number 222:
Narrated Ibn 'Abbas (RA):
The Prophet came to Medina and saw the Jews fasting on the day of Ashura. He asked them about that. They replied, "This is a good day, the day on which Allah rescued Bani Israel from their enemy. So, Moses fasted this day." The Prophet said, "We have more claim over Moses than you." So, the Prophet fasted on that day and ordered (the Muslims) to fast (on that day).
Sahih Bukhari, Volume 3, Book 31, Number 147:
Narrated Salama bin Al-Akwa:
Once the Prophet ordered a person on 'Ashura' (the tenth of Muharram) to announce, "Whoever has eaten, should not eat any more, but fast, and who has not eaten should not eat, but complete his fast (till the end of the day).
Sahih Bukhari, Volume 3, Book 31, Number 117:
Narrated 'Aisha (RA):
(The tribe of) Quraish used to fast on the day of Ashura' in the Pre-lslamic period, and then Allah's Apostle ordered (Muslims) to fast on it till the fasting in the month of Ramadan was prescribed; whereupon the Prophet said, "He who wants to fast (on 'Ashura') may fast, and he who does not want to fast may not fast."
Sahih Bukhari, Volume 3, Book 31, Number 219:
Narrated 'Aisha (RA):
Allah's Apostle ordered (the Muslims) to fast on the day of 'Ashura', and when fasting in the month of Ramadan was prescribed, it became optional for one to fast on that day ('Ashura') or not.