Don't you understand? I want to stay...with...YOU!.
Sara Altney is the princess of Sasune, the kingdom that Final Fantasy III begins in. She is the second in a recurring tradition of Final Fantasy of naming a princess Sara, but was the first one after "Sarah" of Final Fantasy I.
Sara is the first companion of the game, being encountered in the Sealed Cave; Home of the Djinn who was unsealed during the recent earthquake and who has cursed the castle and nearby town of Kazus, turning their people into ghosts.
Just as the original Final Fantasy subverted the "rescue the princess" plot by having it resolved after the first dungeon, Final Fantasy III subverts in another way: The assumption is that Sara was abducted by the Djinn due to her possessing a Mythril Ring: The one thing that can reseal him (indeed, Ingus and the king assume as much in the remake). However, finding her in the Sealed Cave, it turns out to be the opposite situation: Sara is actually enroute to seal the Djinn herself, although she admits that she was nervous prior to the party coming along. Said concern turns out to be founded when the ring alone is not enough to seal the Djinn. Luckily, with the Warriors of Light beating the Djinn into submission, she is finally able to seal him again and Sara breaks the curse by throwing the ring into Sasune's blessed spring.
The princess expresses a desire to accompany the youths on their journey, but fully and sadly acknowledges that she would only slow them down.
The Final Fantasy III remake makes a few expansions regarding Sara. The most immediately obvious is Sara's interactions with Ingus, with it being clear that she has a thing for him. This in particular changes the nature of Sara's introduction: Rather than refusing to go back and trying to go forward before deciding to go with the Warriors of Light, the fact that she is Ingus' superior effectively shuts down the argument immediately.
While saying goodbye to Sara is optional, it's expanded onto a full scene in the remake. When Ingus goes up to her room to say goodbye, Sara reveals that she specifically wanted to avoid this and starts to cry. In one of the few cases in the game, Ingus addresses her simply as "Sara" rather than as "Lady Sara". Regardless of if you say goodbye or no, Sara will appear on the back battlements as the Warriors leave, waving goodbye and wishing them luck.
Like the rest of the guests, Sara can be messaged via Mognet. Most of her letters are standard, with the only standouts being two: Her second letter and her final letter (for starting the Mognet quest). The former plays a bit into her tomboyish side, making up for her changed introduction; By having her express boredom with being stuck at the castle and asking the Warriors to take her on an adventure someday.
Sara kicks off the Mognet sidequest proper by reaching out to the Warriors to have her pendant fixed. This eventually leads the Warriors to find the Legendary Smith, who fixes it. The reason why Sara wanted it fixed so badly? It's a keepsake from her mother. This is the only time in either version of the game in which the Queen of Sasune is brought up.
When Sara aids in battle, she uses Aero and Cure, positioning her as a White Mage.
Despite being very early in the game, Sara actually has a decent chunk of content linked to her.
A rare curious plot point that got dropped: Sara's Mythril Ring was actually created for Refia, though she tries to deny it when Sara tries to press her about it. This isn't entirely followed up on aside from Refia trying to act unconcerned when Sara presses her about it again later only to react with shock when Sara toss it into the spring. Sara assures her that the ring will protect Refia and Refia relents that that's probably what it'd want anyway.
Castle Sasune has numerous traces of unused text and some of it is linked with Sara. In particular, there's text that suggests that Sara and Ingus' parting was once more dramatic and that they would have spent the night together. Additionally, there is dialogue where Ingus admits to Sara that he's ashamed at how he did nothing when the Djinn cursed Sasune, in contrast with Sara who actually went out to fix the problem.