The literal meaning of the Arabic word Aashuraa is “tenth”. In Islam, Aashuraa refers specifically to the 10th day of the lunar month called Muharram of the Muslims calendar. Muharram is the first month of the Islamic calendar. Islamic calendar is pegged to the moon's 29-day or 30-day cycle, and so moves from year to year.
Ibn 'Abbas (may Allah be pleased with him) said: "The Prophet (peace and blessings of Allah be upon him) came to Madinah and saw the Jews fasting on the day of 'Ashura'. He said, 'What is this?' They said, 'This is a righteous day, it is the day when Allah saved the Children of Israel from their enemies, so Musa fasted on this day.' He said, 'We have more right to Musa than you,' so he fasted on that day and commanded [the Muslims] to fast on that day." [Reported by al-Bukhari, 1865] "This is a righteous day" – in a report narrated by Muslim, [the Jews said:] "This is a great day, on which Allah saved Musa and his people, and drowned Pharaoh and his people." "Musa fasted on this day" – a report narrated by Muslim adds: " … in thanks giving to Allah, so we fast on this day." SahihMuslim 2520
Narrated by`Aisha رضي الله عنه : The people used to fast on the day of 'Ashura' before fasting in Ramadan was prescribed but when (the order of compulsory fasting in) Ramadan was revealed, it was up to one to fast on it (i.e. 'Ashura') or not. Sahih Al Bhukari Vol 3: 116, 117 and Vol 6:29
Jabir b Samura reported that the Messenger of Allah (may peace be upon him) commanded us to observe fast on the day of Ashura and exhorted us to do it and was particular about it But when (fasting) in Ramadan was made obligatory, he hence. forth neither commanded us nor forbade us, nor was he so particular about it. Sahih Muslim 1128, 1125a (NE 2499, 2514) 
An-Nawawi (may Allah have mercy on him) said: "'Ashura' and Tasu'a' are two elongated names [the vowels are elongated] as is stated in books on the Arabic language. Our companions said: 'Ashura' is the tenth day of Muharram and Tasu'a' is the ninth day. This is our opinion, and that of the majority of scholars. This is the apparent meaning of the ahadith and is what we understand from the general wording. It is also what is usually understood by scholars of the language." [Al-Majmu'])'Ashura' is an Islamic name that was not known at the time of Jahiliyyah. [Kashshaf al-Qina', part 2, Sawm Muharram]
Ibn Qudamah (may Allah have mercy on him) said: "'Ashura' is the tenth day of Muharram. This is the opinion of Sa'idibn al-Musayyib and al-Hasan. It was what was reported by Ibn 'Abbas, who said: 'The Messenger of Allah (peace and blessings of Allah be upon him) commanded us to fast 'Ashura', the tenth day of Muharram.' [Reported by at-Tirmidhi, who said, a sahih hasan hadith] It was reported that Ibn 'Abbas said: 'The ninth,' and reported that the Prophet (peace and blessings of Allah be upon him) used to fast the ninth. [Reported by Muslim] 'Ata' reported that he said, 'Fast the ninth and the tenth, and do not be like the Jews.' If this is understood, we can say on this basis that it is mustahabb (encouraged) to fast on the ninth and the tenth, for that reason. This is what Ahmad said, and it is the opinion of Ishaq."
Ibn 'Abbas (may Allah be pleased with them both) said: "I never saw the Messenger of Allah (peace and blessings of Allah be upon him) so keen to fast any day and give it priority over any other than this day, the day of 'Ashura', and this month, meaning Ramadhan." Sahih al-Bukhari, 1867
The meaning of his being keen was that he intended to fast on that day in the hope of earning the reward for doing so. The Prophet (peace and blessings of Allah be upon him) said: "For fasting the day of 'Ashura', I hope that Allah will accept it as expiation for the year that went before." Sahih Muslim, 1976 This is from the bounty of Allah towards us: for fasting one day He gives us expiation for the sins of a whole year. And Allah is the Owner of Great Bounty.
Shaykh-ul-Islam Ibn Taymiyyah (May Allah have mercy on him) said: "Taharah, Salah, and fasting in Ramadhan, on the day of 'Arafah and on 'Ashura' expiate for minor sins only." [Al-Fatawa al-Kubra, part 5]