The Umayyad Principality, as you call it, is more typically known as the Umayyad Caliphate or the Umayyad Dynasty. This was a Muslim state that was powerful for almost 100 years starting in the 660s (661 to 750). It was based in Damascus and it purported to rule all Muslims.
The Prophet Muhammad died in the year 632. After he died, his successors became known as caliphs. They were the political and spiritual leaders of all Muslims since Islam, at that time, was still limited to a relatively small geographical area. The second caliph was a man named Umar who died in 644. He was replaced as caliph by Uthman, who was from a wealthy clan called the Umayyads. Uthman was murdered in 656. After a few years of conflict as to who was the true caliph, a member of the Umayyad family named Muawiyah became the first caliph of the Umayyad Caliphate.
This is important in the history of Islam for two reasons. First, the Umayyads ruled the Muslim world for almost 100 years. This made them, to that point, that dynasty that had ruled Islam for longer than any other. Perhaps more importantly from a modern point of view, the ascendance of the Umayyads led to the split between Sunni and Shia Muslims. When Uthman was murdered, Muhammad’s son-in-law, Ali, was acclaimed as the new caliph by some. He was himself murdered, allowing the Umayyad’s to take power. Shiites believe that Ali was the rightful caliph and that the caliphate should have been passed down through his descendants. The disputes between them and the Sunnis continue to be extremely relevant today.
So, the Umayyad Principality (or Caliphate or Dynasty) was a caliphate that claimed political and religious sovereignty over all Muslims. It claimed this power from 661 until it was overthrown by the Abbasids in 750.