Most scholars believe that Ahsura is named as such because of “tenth” of Muharram (ten is translated as “Ashara” in the Arabic language)
Sunni Muslims look at Ashura as “good” while Shia Muslims believe that day to be a day of mourning and sorrow.
Based on the Hadith of Prophet Muhammad (saws), Sunni Muslims celebrate Ashura as the day when Prophet Moses (Moosa) fasted on that day because Allah saved the Israelites from their enemy in Egypt. One of the many Ahadith (sayings of Prophet Muhammad) that attests to that is in Bukhari that states:
Narrated by al-Bukhaari (1865) from Ibn ‘Abbaas, who said: The Prophet (saws) came to Madinah and saw the Jews fasting on the day of Ashoora. He said, “What is this?” They said, “This is a good day, this is the day when Allah saved the Children of Israel from their enemy and Moosa fasted on this day.” He (the Prophet Muhammad) said, “We are closer to Moosa than you.”
So he (the Prophet Muhammad) fasted on this day and told the people to fast.
There are many other versions of this Hadith in the books of “Muslim” and “Bukhari”.
According to a version narrated by Muslim,
This is a great day when Allah saved Moosa (Moses) and his people and drowned Pharaoh and his people.”
Sunni Muslims celebrate Ashura by fasting on that day. Usually, Sunni Muslims are recommended to fast on the 9th and 10th of Muharram.
It is permissible to fast the day of Ashoora on its own, but it is better to fast the day before it or the day after it. This is the Sunnah that is proven from the Prophet (peace and blessings of Allah be upon him), as he said, “If I am still alive next year, I will certainly fast the ninth.” Narrated by Muslim, 1134. [Source: Islam-QA.com (question 21776)]
Other Ahadith on the subject are the following:
The prophet observed the fast on Ashuraa (the 10th of Muharram), and ordered (Muslims) to fast on that day. (Agreed upon Hadith i.e. Bukhari & Muslim).
Narrated by Abi Katada: The prophet was asked about fasting on ‘Ashuraa’ (the 10th of Muharram), he said: “it expiates the previous year (for sins).” (Sahih Muslim)
It was proven from the Prophet (peace and blessings of Allaah be upon him) that the best fasting after Ramadaan is fasting in the month of Muharram. It was narrated that Abu Hurayrah (may Allaah be pleased with him) said: The Messenger of Allaah (peace and blessings of Allah be upon him) said: “The best fasting after Ramadaan is the month of Allaah Muharram, and the best prayer after the obligatory prayer is prayer at night.” Narrated by Muslim, 1163.
Shia Muslims observe Ashura as the day of martyrdom of Hussein ibn Ali, the grandson of Prophet Muhammad at the Battle of Karbala. Shia Muslims therefore consider this a day of sorrow and observe it as such by refraining from music, listening to sorrowful poetic recitations, wearing mourning attire, and refraining from all joyous events (e.g. weddings) that in anyway distract them from the sorrowful remembrance of that day.
Some Shia sects carry the observance to further extremes by beating themselves in public. Sunni Muslims believe such Shia practices (beating, etc.) of the day of Muharram (Ashura) as innovations. Their basis is that during the lifetime of Prophet Muhammad (peace and blessings of Allah be upon him) a number of senior Sahabah (Prophet’s companions) were martyred and he mourned their loss, such as Hamzah ibn Abd al-Muttalib, Zayd ibn Haarithah, Jafar ibn Abi Taalib and Abd-Allah ibn Rawaahah, but he never practiced any such acts. [References: Islam-QA.com]
As 9th of Muharram for the year 1433 Hijri will fall on December 4th, 2011, that day and the day after will be the two days Muslims should be looking to fast.