بسم الله الرحمن الرحيم
The Prophet (pbuh) said: "For fasting the day of ‘Ashoora’ I hope Allah will expiate thereby for the year that came before it." [Muslim]
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The 10th of Muharram is the Day of Ashura, which will be on Saturday inshAllah. The reason for fasting on this day is because it was the day that Allah saved Moosa (as) and the Children of Israel from Pharoah.
It is a sunnah to fast a day on the 9th as well, and if one cannot, then on the 11th. Fasting only on the 10th is also permitted. The Prophet (pbuh) said: "Fast a day before it or a day after." [Baihaqi]
To read more, see here.