The Tenth of Muharram is called Yawmul Ashura. There a many barakah in fasting on this day. In the beginning this fast was fardh [mandatory] on the Muslims. Sayyidina Rasuli Akram Sall Allahu alaihi wa Aalihi wa Sallim ordered to fast on this particular day. When Allahu ta 'ala made fasting the month of Ramadhan fardh, the fast of Muharram became nafl [optional]. Hazrat Ibn 'Abaas Rathi Allahu ta 'ala anh says that Rasulullah Sall Allahu alaihi wa Aalihi wa Sallim said: "There is no superiority of fasting on any particular day over another except the fasting during the month of Ramadhan and on the Tenth of Muharram."
It is Mustahaab [highly meritorious] to fast on the ninth and tenth of Muharram. Some 'Ulama [True Islamic Scholars] have said that fasting only on the Tenth of Muharram is makruh [frowned upon\blameworthy] because it is like the fast of the Yahudi [Jews]. If he fast on the eleventh it is better. One should engage in 'Ibaadah on the night of Ashura. He will receive much thawaab from it. One should recite Dua Ashura on the Tenth of Muharram having taken a ghusl [complete bath rinsing mouth and nose in the process] the day before. He should recite four nawaafil rakaats with one niyyat [intentions]. In every rakaat, after Al Hamd Sharif [Suratul Fatihah] , he should recite Suratul Ikhlaas fifteen times and after taslim [salaams] he should recite this Dua' ten times.
The thaawaab of the salaat [esaale sawaab] should be sent on the arwaah [souls] of Sayyidayn wa Shaahidayn Imaams Hazrat Hasan and Hazrat Hussayn Ridhwaan Allahu ta 'ala alaihim ajma'een. It will bring the reciter a lot of thawaab as well. Because of the reciter having done this, Sayyidayn wa Shahidayn wa Imamayn Hassan and Husayn will do Shifaa [intercede] for him on the Day of Judgement. The adaab of this particular day is to make ghusl, fast and break fast, make salaat, make Dua', visit the sick, be kind to orphans and make peace between to disputants.
One should cry
from fear of Allah, use kohl, give lots of food to the family, put on some attar,
guide people who have gone astray. To recite Qur'an Sharif, to recite Kalima
Tamjeed seventy  times. One should visit the Qabrastan [cemetery], met with
friend and relatives, serve parents and abstain from family relationship with