The Steward of the Dragon is a title given to those people granted rule over a nation by Rand al'Thor when there are not available rightful rulers.
Stewards were assigned in both Tear and Cairhien, Dobraine Taborwin being Steward of the Dragon in Cairhien. The Steward in Tear, Darlin Sisnera, has since been raised to be the first King of Tear.
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.