The Steward of the Dragon is a title given to those people granted rule over a territory or 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 officially assigned. They can be considered as the first unofficial Stewards of the Dragon over Cairhien.
After Dumai's Wells battle the cairhienin noble Dobraine Taborwin was officially appointed as Steward of the Dragon in Cairhien, and he ruled for a quite long period, with the aid of some Aes Sedai and Aiel.
Finally when Dobraine was sent by Rand to Arad Doman the two Aes Sedai Sashalle Anderly and Samitsu Tamagowa took unofficially the stewardship of Cairhien for a while, until the coming of Elayne as new queen.
Other more brief or particular cases of stewardship
Caemlyn and Andor
After the conquest of Caemlyn against Rahvin, Rand proclaimed that the legitimate ruler was Elayne and that he was waiting for her return. During his frequent absences from Caemlyn, instead than appointing an official Steward Rand left the rule to an informal stewardship council leaded by Lord Bashere, Bael and Melaine.
Initially Rand transferred Lord Dobrain from Cairhien to Bandar Eban as his Steward, but the situation worsened and soon the forces of the Dragon were forced to abandon the city. After his second return to Arad Doman Rand appointed Iralin  as his Steward in the city of Bandar Heban.
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.
Steward figures have been common in human history  and they might best be compared to the governors that controlled Imperial Territories in the Roman Empire.