The most virtuous day of Muharram (the First Month of Islamic Calendar) is the day of Ashura - the tenth of Muharram. It is reported in Sahih al-Bukhari, when Allah's Messenger (peace be upon him) came to Madinah, he saw the Jews fasting on the day of Ashura. He said: 'What is this?' They said: 'This is the day when Allah saved Musa from drowning, so we fast on this day.' Then he said: “We have more right to Musa than you.” So he fasted on this day and advised the Muslims to do the same.
Also, Ibn Abbas (may Allah be pleased with him) narrated: “When Allah's Messenger observed the fast of the day of Ashura and ordered (his Companions) to fast, they said: “O Messenger of Allah! It is a day revered by Jews and Christians.” Allah's Messenger said: “The coming year, if Allah wills, we will fast on the ninth (also).” [Sahih Muslim, vol: 1, p: 359]
Ibn Abbas also reported: “I never saw the Prophet more keen to fast on any day other than the day of Ashura and any month than the month of Ramadan.” [Sahih al-Bukhari]
The phrase 'more keen' of Ibn Abbas indicates that the Prophet looked forward to fast on the day of Ashura in order to earn the reward for doing so. The Prophet said: “Fasting the day of Ashura, I hope will be an expiation of sins for the year before.” [Sahih Muslim v: 1, p: 368]
Thus, the day of Ashura is mercy from Allah and an opportunity for one to get his sins forgiven. It is recommended for the Muslim Ummah to fast both the tenth and the ninth of Muharram since Allah's Messenger fasted the tenth and intended to fast the ninth. Another important thing to be understood here is that voluntary fasts are expiation for minor sins, as long as one completely refrains from major sins and is always cautious about minor sins. Some people who are deceived rely greatly on voluntary worships (like fasting on the day of Ashura or the day of Arafat) while neglecting the obligatory worships (praying five daily prayers, fasting in Ramadan, etc.)
"Muharram: The Month of Allah" - As-Sunnah Bimonthly Islamic Newsletter