In the Arabic Society, ashura was esteemed even before Islam; since the time of Prophet Abraham.
Regarding the precious blessings and events that Allah bestowed on this day, till the Ramadan fast became compulsory, Prophet Muhammad (pbuh) gave a great importance to it and never abandoned Ashura fast.
Aisha (ra) narrated that:
“Quraish used to fast on the day of 'Ashura' in the Pre-lslamic period, and Allah's Apostle too, used to fast on that day. When he came to Medina, he fasted on that day and ordered others to fast, too. Later when the fasting of the month of Ramadan was prescribed, he gave up fasting on the day of 'Ashura' and it became optional for one to fast on it or not.” (Bukhari, Volume 3, Book 31, Number 220)
In Islam, it is not obligatory to fast on the Day of Ashura. It is “sunnah” of Our Prophet (pbuh), He fasted and recommended fasting; but not in the way that the Jews did. The Jews used to fast only on the Day of Ashura, which is the 10th day of Muharram. So, in order not to be similar to them, He recommended fasting for two or three days; on the both 9th and 10th of Muharram or both the 10th and 11th of Muharram.
Another reason that makes Ashura remarkable in Islam is that it is the day when Husayn (ra), the blessed grandson of Prophet Muhammad and the master of the youth of heaven, was martyrized. Husayn (ra) and his companions opposed Yazid’s tyranny and in consequence, by the command of Yazid, they were martyrized atrociously in Karbala on the 10th of Muharram in Hijri 61 (10 October 680)