Not related to Ashura and Karbala, some Sunni Muslims fast on this day of Ashura based on narrations attributed to Muhammad. Some other Sunnis accept Ashura as a significant day due to the martyrdom of Imam Husayn and the significance of the events at Karbala. The fasting is to commemorate the day when Moses and his followers were saved from Pharaoh by Allah by creating a path in the Red Sea. According to Muslim tradition, the Jews used to fast on the tenth day. So Muhammad recommended to be different from the Jews and recommended fasting two days instead of one. This quote has been said many times but there is no valid hadees from any authentic source. 9th and 10th or the 10th and 11th day of Muharram.
In some countries other religious communities commemorate this event. According to Hadith record in Sahih Bukhari, Ashura was already known as a commemorative day during which some Makkah residents used to observe customary fasting. Muhammad used to fast on the day of Ashura, 10th Muharram, in Makkah. When fasting the month of Ramadan became obligatory, the fast of Ashura was made non compulsory. This has been narrated by Ayesha RA, Sahih Muslim, (Hadith-2499). In hijrah event when Muhammad led his followers to Madina, he found the Jews of that area likewise observing fasts on the day of Ashura. At this, Muhammad affirmed the Islamic claim to the fast, and from then the Muslims have fasted on combinations of two or three consecutive days including the 10th of Muharram (e.g. 9th and 10th or 10th and 11th).
A companion of Muhammad, Ibn Abbas reports Muhammad went to Madina and found the Jews fasting on the tenth of Muharram. Muhammad inquired of them, "What is the significance of this day on which you fast?" They replied, "This is a good day, the day on which God rescued the children of Israel from their enemy. So, Moses fasted this day." Muhammad said, "We have more claim over Moses than you." Muhammad then fasted on that day and ordered Muslims too.
The narrations of Muhammad mentioning the Children of Israel being saved from Pharaoh are indeed confirmed by authentic hadith in Sahih Bukhari and Sahih Muslim.

Sunnis regard fasting during Ashura as recommended, though not obligatory, having been superseded by the Ramadan fast.Sahih Muslim, (Hadith-2499)

Muhammad's tribe, the Quraish, fasted on the 10th of Muharram. Though optional, Muhammad retained this pre-Islamic practice too. Below is details from the Hadith:

Narrated Ayesha RA:

'Ashura' (i.e. the tenth day of Muharram) was a day on which the tribe of Quraish used to fast in the pre-lslamic period of ignorance. The Prophet also used to fast on this day. So when he migrated to Madina, he fasted on it and ordered (the Muslims) to fast on it. When the fasting of Ramadan was enjoined, it became optional for the people to fast or not to fast on the day of Ashura.

Egyptian Muslims customarily eat a pudding (also known as Ashura) after dinner on the Day of Ashura. Similar to the Turkish Aşure, it is a wheat pudding with nuts, raisins, and rose water.