Q: What is the ruling on fasting in Muharram? Should I fast the 9th and the 10th?
A: It is recommended to fast Allah's month, al-Muharram due to the hadith, "The most virtuous prayer after the those obligated is prayer at night and the most virtous fast after Ramadan is Allah's month al-Muharram". [Muslim]
It is possible that the Prophet (peace be upon him) did not fast alot during the month due to some excuse or him not knowing of its virtue until the end of his life.
Ibn al-Athir noted that Allah's ascription of this month is done out of veneration and glorification much like when He says the House of Allah or the People of Allah.
The most stressed, some say the most virtuous, is the 10th which is called Ashura. During that day it's recommended to spend on the family as has been said in al-Mubdi'.
Fasting on Ashura is an expiation of sins for the year based on the hadith, "I take account with Allah that it will expiate the previous year".
After Ashura, the most stressed is the 9th which is called Tasu'a due to the hadith reported by Ibn Abbas, "If I am alive next year, I will certainly fast the 9th and 10th". [al-Khallal and used as evidence by Imam Ahmad]
source: Sharh al-Muntaha