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 a day of “respect and gratitude” (for Prophet Moosa and his nation), while Shia Muslims believe that day to be a day of mourning and sorrow. The following is an explanation of the difference.
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-Bukhari (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 (Prophet Moses) than you.”
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.
Al-Shafa’i and his companions, Ahmad, Ishaaq and others said: It is mustahabb [recommended] to fast both the ninth and the tenth, because the Prophet (peace and blessings of Allah be upon him) fasted the tenth and intended to fast the ninth. Based on this, there are different ways of fasting ‘Ashoora’, the least of which is to fast the tenth only, but it is better to fast the ninth as well. The more one fasts in Muharram, the better. (islamqa.info)
In Jewish tradition, this festival is celebrated as the Passover, which is their way to commemorate their liberation over 3,300 years ago by God from slavery in ancient Egypt that was ruled by the Pharaohs, and their freedom under the leadership of Prophet Moses. It commemorates the story of the Exodus as described in the Hebrew Bible especially in the Book of Exodus, in which the Israelites were freed from slavery in Egypt.
Other Ahadith on the subject are the following:
The prophet observed the fast on Ashura (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 Ashura (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 Allah be upon him) that the best fasting after Ramadan is fasting in the month of Muharram. It was narrated that Abu Hurayrah (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “The best fasting after Ramadan is the month of Allah, Muharram, and the best prayer after the obligatory prayer is prayer at night.” Narrated by Muslim, 1163.
Shia Muslims Observance
Shia Muslims’ observance of Ashura is different altogether. They 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. (Read here about Sunni Shia History and Conflicts) Also, see the comments below and add yours with facts.