Dragonsworn is the name given to a group of people who have sworn to serve and fight for the Dragon Reborn. The term is applied whether the Dragon they have sworn to follow is Rand al'Thor or a False Dragon like Taim or Logain.
Often times the Dragonsworn are no more than rioting peasants. Many of these groups have never had any political contact with the Dragon and frequently cause pandemonium in the land in misguided loyalty. As a result they clash frequently with the armed forces of whatever nation they arise in.
It should be noted that while the term "Dragonsworn" is applicable to anyone sworn to the Dragon, it is generally used only towards groups by those in disapproval.