Her expertise in archaeology, anthropology and ancient civilizations made her a useful part of the landing party -- the crew was confronted by an alien of unknown name and purpose, but who seemed familiar with the Earth of 5,000 years ago. This alien soon identified himself as Apollo, and claimed to be the being ancient Greeks believed was a god.
Apollo was moved by Palamas' beauty, and quickly became infatuated with her. He transformed her standard uniform into a long peplos, which he felt was more appropriate attire for his future queen. On several occasions, he left the remainder of the landing party alone, taking the young lady elsewhere to entertain her. This created tension between the alien and Scotty, who was also interested in Palamas; on two occasions, this drove Scotty to attack Apollo, with unfortunate results for the ship's chief engineer. And, she began to develop feelings for Apollo, in return.
Finally, out of options, Kirk confronted Palamas, and demanded that she reject Apollo. Kirk knew Apollo was desperately lonely, and knew also that Apollo thrived on attention and worship -- especially Carolyn Palamas' attention. At first, she refused, claiming she loved the alien. Kirk forced her to confront an ugly truth: although Kirk was fine with Apollo being Carolyn's love interest the lives of everyone, on the planet and on the ship, depended on her remembering her duty. She became uncertain; struggling to balance her view of Apollo as a kind and paternal benefactor with Kirk's view of Apollo as slave-master. Ultimately, Kirk reminded her that she was Human, and renewed her connection -- and duty -- to the rest of Humanity, in the form of her shipmates and friends aboard the Enterprise. Additionally, the Captain implied that once Apollo achieved his ends (making her the mother of the thousands of gods he wanted), he would have no more need of her and kill her.
Heavy-hearted, Palamas left to meet with Apollo at his call, and on her captain's orders she rejected him, pretending that her entire association with him was part of her study of alien civilizations. Apollo was first confused, hurt, and then enraged. He expended a great deal of power to impress her, summoning a storm and crafting a gigantic image of himself in the sky. Kirk's implications for Carolyn's life and safety prove to be right, as Apollo uses the storm he summons to scar Carolyn's face and dishevel her hair.
About this time, Spock and the Enterprise crew managed to pierce Apollo's hand in places, and they attacked his power source through these holes. Apollo, having expended much of his energy in his rage, was unable to prevent the destruction of his temple. Without it, he was powerless, and he "spread himself on the winds", joining his old companions in oblivion, an event that left Palamas in tears. (TOS: "Who Mourns for Adonais?")
Background information Edit
Palamas was played by actress Leslie Parrish.
The original teleplay, and James Blish's adaptation, contained an additional event that did not make the filmed episode: at the end of the story, Carolyn Palamas is revealed to be pregnant with Apollo's child. According to the Star Trek Encyclopedia (4th ed., vol. 2, p. 121), an early draft script contained this revelation, as well. In the novelization the child is a daughter, Carolyn would name Athena. Also, in the original script the character is named Carolyn Bassett.  The scene which divulged Palomas' pregnancy was actually filmed, but omitted from the episode as aired. Recently rediscovered, the outtake is featured in the "Inside the Roddenberry Vault (Part 3)" special feature on the 2016 Star Trek: The Original Series - The Roddenberry Vault Blu-ray Disc-set.
Peter David's New Frontier character Mark McHenry was a descendant of this child, and possessed at least some of these beings' powers. Palamas also has a major role in the Judith and Garfield Reeves-Stevens novel Prime Directive.
Palamas and Scotty's first meeting is detailed in Dave Galanter's novel Crisis of Consciousness. After Scott breaks his wrist and is unable to work, she keeps him company while Kirk fetches a medkit. Trained as a field medic, she is able to heal him enough to allow him to return to his duties, much to Scott's relief. The novel also shows her in her role as archaeology and anthropology officer when dealing with the Maabas (β).
According to the video game Star Trek: Starship Creator, her parents are named Leslie and Robert Palamas. She also has a brother named Mark and a sister named Ellen.