Trespassers! Don't you know this is the hall of the Great Mage Doga?
Doga is a Mage who lives a secluded life in a manor deep in the Dalg continent. With the fierce winds preventing any old airship from approaching, it's not until the Warriors of Light gain the Nautilus from Saronia that they can reach him, as they are told by the mages of Saronia. Upon entering the manor, a voice calls them trespassers before unleashing upon them... a cadre of cute fluffy creatures. Doga then arrives, realizes who his guests are and speaks to them.
Doga is something of a turning point for FF3's plotline; Up to this point, the Warriors of Light have just been kinda wandering around with the only inkling to the greater plot being that someone named "Xande" is sending monsters to cause chaos. Doga explains who Xande is; Xande, Doga and Unei were all students under a great sage named Noah. For centuries, the three studied under him and, as he was dying, he gifted each of his apprentices a special gift: Doga got great magical prowess, Unei got mastery over the realm of dreams and Xande got the gift of mortality.
After explaining the plot to the Warriors of Light, Doga joins them in the Cave of the Circle; A cave leading from Doga's Manor to a circle that connects to the World of Darkness. After reaching the circle, Doga enchants the Nautilus so that it can go underwater and sends the party off to find Unei while he enters the rift between dimensions to search for the key to Eureka.
After the party finds Unei and the Fang of Earth, Doga teleports them to a grotto where he and Unei await. To complete the key to Eureka, Doga and Unei force the party to fight them to the death.
The Warriors of Light are...less than thrilled to do so.
Despite this, The heroes succeed and Doga hands them the finished key to Eureka, reassuring them that the old folks sprits will remain.
And they're right. When the Warriors of Light are caught in Xande's trap, Doga goes around to each of the friends that the Warriors of Light made and summons them to their aid.
Alas, they can't keep their word forever. As the Warriors of Light lie dying due to the Cloud of Darkness' attack, Doga and Unei appear one last time. They share their souls with the Warriors of Light, reviving them but severing the tether they had to the world. Thus, they ascend to become one with the Great Spirit, encouraging the Warriors of Light to go after the Cloud of Darkness.
Doga is a "wise old man" character through and through. Doga doesn't directly display any great magical prowess until his boss fight, but he does mention that the crystal Goldor had was actually a totally different one from the elemental crystals, one he produced to make gold. Additionally, an optional area is Doga's Village, where mages quietly study the forbidden magics in peace.
Interestingly, despite being immortal, Doga remains something of a frail old man, having a bad cough and mentioning that he doesn't think he'll last much longer. The remake kinda explains this and the Crystal Tower sequence hints at said explantion, but still odd.
Doga's Manor is largely staffed by Moogles, who explain some things that Doga himself does not, such as how Xande left the manor annoyed after receiving his gift and how the Cave of the Circle is linked to the World of Darkness.
The remake doesn't expand on Doga that much compared to the other guests, at least in the final game. His biggest difference is mentioning the virtues of each of the warriors (their lights in English, their hearts in Japanese).
In battle, Doga primarily uses Black Magic spells, notably Firaga and Flare.
Doga's more substantial differences lie in the unused text. There, Doga talks of a number of topics, such as musing on balance between light and dark and whoever (if it was indeed a person) decided upon it and the nature of the world and the sun (a topic that made it into the final game in the Warriors of Darkness' speeches, but is more prominent here). Interestingly, Doga notes that the Warriors of Darkness are said to have died fighting against the Wrath of Light. Luneth is confident that the Warriors of Light won't die so easily, however.
A major theme running through these scenes is that Ingus is deeply suspicious of Doga. Originally, he believes Doga to be in cahoots with Xande; For his part, Doga doesn't chastise him and even gives the party explicit permission to kill him if they think he's up to no good (although Ingus argues he woulld've done so regardless). While Ingus never levels any accusations on that level, he does believe that Doga is trying to rope the party into a fight with Xande and generally remains suspicious throughout the Cave of the Circle. Doga, on the other hand, shows some perception when he mentions that Ingus' eyes are similar to those of a Dark Blade wielder.
On the world map, the party has some dialogue for after their encounter with Doga. As Arc lays blame on him and Luneth for preventing Doga from telling them more, Ingus mentions that he felt uneasy with how Doga was testing them. Refia collaborates his feelings, comparing it to whenever she felt like rebelling against her father. When Ingus elaborates that he didn't feel he could fully trust Doga, Luneth contrasts him with Elder Topapa, who would usually lecture him while Arc is confident that Doga trusts them. The theme of Doga and Unei testing the four is a recurring one through these sections of the unused text, with the four finally feeling that they passed after receiving the keys.