The Irish Dragoons were a regiment named under the understanding of the job of Heavy Cavalry in the 18th Century. However, Dragoons had been in existence prior to that date and the term in the song could refer to a group of Dragoons (basically mounted infantry who happened to be Irish as opposed to a specific regiment of Irish Dragoons.   

During the English Civil War dragoons were used for a variety of tasks: providing outposts, holding defiles or bridges in the front or rear of the main army, lining hedges or holding enclosures, and providing dismounted musketeers to support regular cavalry. Supplied with inferior horses and more basic equipment, the dragoon regiments were cheaper to recruit and maintain than the expensive regiments of cavalry.