Ashura is the tenth day of the month of Muharram. This is the opinion of the majority of scholars and is the preferred point of view. Ashura was a day on which the tribe of Quraish used to fast in the pre-Islamic period of ignorance. The Prophet (PBUH) also used to fast on this day. So when he (PBUH) migrated to Medina, he fasted on it and ordered (the Muslims) to fast on it. When the fasting of Ramadan was enjoined, it became optional for the people to fast or not to fast on the day of Ashura.
The Prophet (PBUH) said: "One who fasted on the day of Ashura, atoned for equivalent to sixty months or got rewarded to get ten believers released for life from the progeny of Ishmael peace be upon him." [Ar-Rabee'#308]
Ashura commemorates the day that Allah saved the Children of Israel from Pharaoh; and for this reason the Christians and Jew honor this day. Ibn Abbas narrated: "The Messenger of Allah (PBUH) fasted on the day of Ashura and ordered the people to fast on it. They said: 'O Messenger of Allah, it is a day that the Jews and Christians honor.' The Prophet (PBUH) said: 'If I remain (alive) till the next year, I shall (also) observe fast on the ninth (of Muharram).' The narrator said: "The death of the Prophet came before the following year." [Muslim #1916&1917]
Ibn Abbas narrated: "The Prophet (PBUH) 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 the Children of Israel from their enemy. So, Moses fasted this day." The Prophet (PBUH) 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)." [Al-Bukhari#1865]
The scholars have mentioned that the fast of Ashura is of three levels:
1- fasting three days; that is, on the 9th, 10th, and 11th of Muharram;
2- fasting on the 9th and 10th; and
3- fasting only on the 10th.
And the best is level 2, i.e. fasting on the 9th and 10th.
The Approved in the Jurisprudence of Zakat and Sawm, by Al-Mu'tasam Al-Ma'awali