It has been authentically reported that the Prophet (sallallaahu 'alayhi wa sallam) fasted the day of 'Aashoora and encouraged the people to do so as that was the day when Allaah saved Moosa (AS) and his followers and destroyed Fir'aun and his followers. It is therefore recommended for every Muslim to fast this day as a sign of gratitude to Allaah. The tenth day of al-Muharram is the day of 'Aashoora (coinciding with Sunday 27th December this year).
It is also recommended that one fasts a day before or after the day of 'Aashoora to be different from the Jews on that day. And if one were to fast all the three days (9th, 10th and 11th of al-Muharram) then this is fine as it has been reported from the Prophet (sallallaahu 'alayhi wa sallam) that he said "Oppose/be different from the Jews, fast a day before it and a day after it." In another narration "Fast a day before it or a day after it."
When the Prophet (sallallaahu 'alayhi wa sallam) was asked about fasting the day of 'Aashoora, he said "Allaah expiates (through it) the sins of the previous year."
There are many hadeeths concerning the fasting of 'Aashoora and encouragement of the same. We ask Allaah to guide us to all that pleases Him and to place us amongst those who race to do every good.
‘Aashooraa’: Celebration or Mourning?
Ruling on celebrating ‘Aashooraa’ or taking it as a day of mourning
See also
http://www.islamqa.com/en/ref/21290
