A skill dome was a standard form of education on Vulcan, including the Vulcan Learning Center. The student was placed in a sunken pit which projected holographic images with a voice asking questions.
Sometime in the 2240s of the alternate reality, a young Spock answered math and philosophy questions while inside a dome pit at the Vulcan Learning Center. It was there where he was often harassed by a trio of Vulcan bullies because of his Human heritage. After thirty-five attempts, they successfully elicited an emotional response from Spock by calling his father a traitor and his mother a "Human whore". In a rage, Spock pushed one of them into a dome pit and physically assaulted him. (Star Trek)
Michael Burnham also attended the Vulcan Learning Center and had to answer questions about Klingons in her dome, which was traumatic for her given that Klingons killed her family. At some point after Burnham began studying at the Center, a bombing devastated the facility. (DIS: "The Vulcan Hello", "Battle at the Binary Stars")
The domes were not named in dialogue, their name coming from the Star Trek script.