Ashura is the term used to signify fasting on the 10th day of the Islamic month of Muharram. It is reported by Sahih Muslim that the Prophet (SAWW) said regarding this day,
Fasting the day of ‘Arafah I hope Allaah will expiate thereby for the year before it and the year after it and fasting the day of ‘Ashoora’ I hope Allaah will expiate thereby for the year that came before it.
As Muslims, the Prophet (SAWW) is our role model and we do our best to emulate his practices. It was the consistent practice of the Prophet (SAWW) to fast on the 10th of Muharram. It was narrated that Ibn ‘Abbaas (may Allaah be pleased with him) said:
I never saw the Prophet (peace and blessings of Allaah be upon him) so keen to make sure he fasted any day and preferring it over another except this day, the day of ‘Ashoora’, and this month – meaning Ramadaan.Narrated by al-Bukhaari, 
Why to Fast on Ashura???
In terms of the significance of this day, fasting on Ashura, it is narrated by Imam Bukhari that Ibn Abbas (RA) said,
The Prophet (peace and blessings of Allaah be upon him) came to Madeenah 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 Allaah 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.
Fasting on Ashura thus expiates minor sins for the previous year. It is to be noted that significant crimes need separate repentance.
In regards to fasting on the 9th or 11th day of Muharram, Imam At-Tirmidhi reported that Ibn `Abbas (may Allah be pleased with them both) used to say: We should fast on two days: the 9th and 10th of Muharram to distinguish ourselves from the Jewish community. (At-Tirmidhi, Hadith 686). The Prophet (SAWW) was reported to have said that he intended to fast on both the 9th and 10th of Muharram to distinguish themselves from the Jewish community. However, the Prophet (SAWW) passed away that year, so his Companions (RA) decided to implement fasting on the 9th and the 10th of Muharram. The scholars later decreed that the Ashura fast should be preceded by fasting on the 9th or followed by fasting on the 11th of Muharram, if the individual can fast those two days. If an individual is unable to fast both days, then she can fast the day of Ashura as there is the great reward in it and should not be left.
May Allah (SWT) grant us the ability to fast Ashura and accept it from us. Ameen