Fasting on the Day of Ashura (10th Muharram) for forgiveness for the previous year’s sins:
The Prophet (Sallallahu Alaihi Wasallam) said: “Fasting on the day of ‘Ashura’ I hope Allah will expiate thereby for the year that came before it.” (forgiveness for the previous years sins) [Muslim, 1162]
The Prophet (Sallallahu Alaihi Wasallam) said, “Next year, if Allah wills, we will fast on the ninth day.” But by the following year he had passed away.
So it is recommended to fast on the 9th as well as 10th or 10th and 11th Muharram.
Ibn Hajar (Ra) said this is to differ from the Jews who also venerate this day.
9th Muharram – Fri 29th September
10th Muharram – Sat 30th September
11th Muharram – Sun 1st October