Fasting on Muharram 10, known as the Day of 'Ashura', expiates for the sins of the past year.
When the Prophet (peace and blessings be upon him) arrived in Madinah in 622 CE, he found that the Jews there fasted on Muharram 10 and asked them the reason for their fasting on this day. They said," This is a blessed day. On this day Allah saved the Children of Israel from their enemy (in Egypt) and so Prophet Musa [Moses] fasted on this day giving thanks to Allah."
The Prophet (peace and blessings be upon him) said, "We are closer to Musa than you are."
He fasted on that day and commanded Muslims to fast on this day. (Al-Bukhari) The following year, Allah commanded the Muslims to fast the month of Ramadan, and the fasting of 'Ashura' became optional. It is also reported that the Prophet (peace and blessings be upon him) intended to fast on the ninth and tenth.
Ibn 'Abbas reported: The Messenger of Allah fasted on the day of 'Ashura' and ordered the people to fast on it. The people said, "O Messenger of Allah, it is a day that the Jews and Christians honor." The Prophet said, "When the following year comes, Allah willing, we shall fast on the ninth." The death of the Prophet came before the following year. (Muslim and Abu Dawud)
Thus, according to scholars, you may choose to fast 'Ashura' on three days (ninth, tenth, and eleventh); two days (ninth and tenth); or one day only (the tenth).
The Prophet (peace and blessings be upon him) said, "Fasting the day of 'Ashura' (is of great merits), I hope that Allah will accept it as an expiation for (the sins committed in) the previous year." (Muslim) But this expiation of minor sins comes only if you avoid major sins. It is a big mistake to rely on fasting a single day to "wipe your slate clean." If you neglect your daily Prayers or the fast of Ramadan, or if you backbite, lie, commit adultery or other major sins, fasting on 'Ashura' alone will not atone for your sins.
The great scholar Ibn Al-Qayyim Al-Jawziyya (b. 691 AH/1292 CE) wrote: This misguided person does not know that fasting in Ramadan and praying five times a day are much more important than fasting on the Day of Arafah and the Day of 'Ashura', and that they expiate for the sins between one Ramadan and the next, or between one Friday and the next, so long as one avoids major sins. But they cannot expiate for minor sins unless one also avoids major sins; when the two things are put together, they have the strength to expiate for minor sins. Among those deceived people may be one who thinks that his good deeds are more than his sins, because he does not pay attention to his bad deeds or check on his sins, but if he does a good deed he remembers it and relies on it.
This is like the one who seeks Allah's forgiveness with his tongue (that is, by words only), and glorifies Allah by saying "subhan Allah" one hundred times a day, then he backbites about the Muslims and slanders their honor, and speaks all day long about things that are not pleasing to Allah. This person is always thinking about the virtues of his saying "subhan Allah" and saying "la ilaha illa Allah" but he pays no attention to what has been reported concerning those who backbite, tell lies, and slander others, or commit other sins of the tongue. They are completely deceived. (Al-Mawsu'ah Al-Fiqhiyyah, part 31, Ghuroor)
You may hear of some customs associated with 'Ashura' such as cooking and eating certain foods on that day, wearing kohl or henna, or even merrymaking.
Others consider this a day of mourning and wailing in honor of the Prophet's grandson Al-Husayn (May Allah be pleased with him), who was martyred in battle on that day. The scholar Ibn Taymiyah (b. 661 AH/1263 CE) stated that all of these are bid'ahs (reprehensible innovations) that should be avoided.
Take advantage of this opportunity to bring yourself closer to Allah on 'Ashura' by fasting and praying for forgiveness.