Ibn Abbas (radhiAllaahu anhu) reports that the Messenger of Allaah (peace and blessings of Allaah be upon him) arrived in Madinah and found the Jews observing fast on the day of ‘Ashura.They said, "It is a day of great significance when Allaah (azz wa jal) delivered Musa (peace be upon him) and his people, and drowned Pharoah and his people. Musa (peace be upon him) observed fast out of gratitude. And we also observe it." The Prophet (peace be upon him) responded, "We have more right, and we have a closer connection with Musa ('Alayhis-salaam) than you have"; so Allah’s Messenger (peace and blessings of Allaah be upon him) observed fast (on the day of ‘Ashura) and gave us orders to observe it.
--Sahih Bukhari & Muslim
The Messenger of Allaah (peace and blessings of Allaah be upon him) was asked about observing a fast on the 10th day of Muharram, and he replied, "It is an expiation for the sins of the preceding year.''
--Muslim
The Prophet (peace be upon him) recommended that we fast on this day, by his Sunnah. He also indicated how we should observe the fast of ‘Ashura.
Hakam ibn Al-Arat (radhiAllaahu anhu) relates:
I went to Ibn Abbas (radhiAllaahu anhu). I said to him,"Tell me about fasting on ‘Ashura. He said, "When you see the new moon of Muharram count the (days) and (begin to) observe fast on the 9th." I said to him, "Is it how the Prophet (peace be upon him) observed the fast?" He said, "Yes."
--Sahih Muslim
Ibn Abbas (radhAllaahu anhu) relates that when the Messenger (peace be upon him) said:
"If I survive till next year, I will definitely observe fast on the 9th of Muharram (as well)."
--Sahih Muslim
The Prophet (peace be upon him) said:
"Observe the fast of 'Ashura and oppose the Jews. Fast a day before it or a day after."
--Baihaqi
Ibn `Abbas (may Allah be pleased with him) used to say:
We should fast on two days: the 9th and 10th of Muharram to distinguish ourselves from the Jewish community.
--At-Tirmidhi
These hadeeth indicate clearly that one should fast on the 9th AND 10th of Muharram. However, if one does not manage to fast on the 9th, then he/she should fast on the 11th of Muharram instead.