The Steward of the Dragon is a title given to those people granted rule over a nation by Rand al'Thor, while there were not available rightful rulers.
Since the liberation of Cairhien from the Shaido, the Dragon choose to support Elayne as the legitimate ruler, but meanwhile Berelain sur Paendrag Paeron and Rhuarc ruled together and briefly the kingdom, before Dobraine was assigned. They can be considered as the first unofficial Stewards of the Dragon over Cairhien.
Elayne Trakand, Queen of Andor, appointed Perrin Aybara as High Lord and Steward of the Dragon in the Two Rivers to justify Perrin's de facto lordship over that region in front of the Andoran nobility. Perrin in turn appointed Tam al'Thor as Lord and Steward during his absence.
Stewards might best be compared to the governors that controlled Imperial Territories in the Roman Empire.