It is recommended, in the Sunni and Zaydi Shi`i schools, to fast the day. There are numerous sahih hadiths about its fasting, some of them mentioning that it was an obligatory fast in the days before fasting Ramadan became obligatory.
وقال أحمد بن عيسى بن زيد بن علي (عليهم السلام) في أماليه: صوم الأيام البيض، ورجب وشعبان، والإثنين والخميس حسن جميل، وجاء فيه فضل كثير، وكذلك يوم عاشوراء
According to one sahih hadith, the Prophet Muhammad later expressed his wish to also fast on the 9th of Muharram (the day before Ashura), but he died before that could happen.
وعن أبي عبد الله عليه السلام عن أبيه، أن عليا عليه السلام قال: صوموا العاشوراء التاسع والعاشر، فإنه يكفر ذنوب سنة (وسائل الشيعة)
A hadith of disputed reliability says that the Prophet recommended, along with the fast of Ashura, fasting either the day before or the day after.
One version (with a weak isnad) indicates all three days.
The dominant view among Twelver Shi`ah is that it is not recommended to fast Ashura. One narration in the Twelver hadith books suggests that it is a fast that was abandoned after Ramadan became obligatory. Other narrations (in Twelver Shi`i books) condemn its continued fasting as a practice initiated by the killers of the Prophet’s grandson al-Husayn, their purpose being to give thanks for his death.
The killing of the Prophet’s grandson was a tragedy, and is a source of grief for Sunnis and Shi`ah. If you fast Ashura, it should definitely not be with the intention of celebrating Imam Husayn’s murder. I hope we can at least start understanding each other better, and realize what we have in common, even if we don’t end up agreeing on everything. And I am pretty sure the Sunnis and Zaydis who fast Ashura are not doing it to celebrate the killing of al-Husayn.