Zenna Henderson's The People series (first published 1952) had advanced human-like aliens with psi abilities settling on Earth. Alexander H. Key's Escape to Witch Mountain (1968) and the later Disney movies using the book as inspiration have the similar plotline of psychic children from a more advanced planet.
Mark Philips in the early 1960s had an FBI agent slowly acquire psi abilities, first learning telepathy before picking up teleportation. In the "Psi-Power" series one could both be born with psi abilities as well as learn them. Anne McCaffrey's Talents universe (first story in 1959) and even her Dragonriders universe (first story 1967) supposed that humans in the future would have psi powers. James Schmitz's "Agent of Vega" (1949) and his Hub Universe stories have humans in the far future with psi powers teamed up with aliens with the same abilities.
Redheads with psi abilities in stories were common a long time before Jean Grey in The X-Men and Marion Zimmer Bradley's Darkover series. That trope came about because of the stereotype of the Celtic people (Irish and Scottish primarily) having "second sight". All the stories I recall reading in the 1960s and 70s which had redheaded psychics usually mentioned the character's Irish or Scottish background.
Movies such as The Power (1968), The Fury (1978) and Firestarter (1984, based on the 1980 book by Stephen King) usually had a government agency studying the phenomenon in a scientific way before things go badly. Television series such as The Sixth Sense (1972) and Beyond Reality (1991-1993) had academics (sometimes at a university) studying psi powers and usually mentioning real life J B Rhine at Duke University who had tests for extrasensory perception. The Omega Factor (1979) television series went back to the government agency trope as did the recent remake of The Tomorrow People.
In real life, J B Rhine at Duke University (who coined the phrase "parapsychology") did have tests for clairvoyance, precognition and telepathy. There were rumors (and books written) of both the Soviets and the Americans having secret agencies using psychic abilities such as remote viewing to see hidden military facilities.
The list of psychic abilities has many different terms, but the ones commonly used in fiction are telepathy, telekinesis, teleportation, clairvoyance and precognition. Even before the overall term of "parapsychology" shifted over to "paranormal", mediums and communication with the dead was a trait that was felt to be more fantasy than science, but how well it was explained could shift the story into one or the other genre. In real life, psychics and mediums are still on television, but drama shows with psychics helping police departments have disappeared. Psychic telephone networks moved to the very very late night and seem to have faded from view. The Wikipedia entry for psychic mentioned a study in 2008 where neurological imaging was done to detect telepathy (which failed) and analyses (latest one done in 2011) of television mediums and how their shows are edited.
But back to ESP and psychic abilities in fiction. Various psychic abilities have been present in fantasy for a long time. The genres of "paranormal" and "urban fantasy" usually had those abilities labeled more as magic. Clairvoyance or precognition might require a crystal ball for focus, levitation may or may not need a wand, and teleportation (or apportation) would use a spell or magic portal. The terms used were not always the same between the genre, which was one way to distinguish them. Whether the story was categorized as fantasy or science fiction usually depended on the actual story itself and not just the inclusion of one element. I write stories with characters with psychic abilities. Some stories are fantasy with elves and wizards and an empath who can heal and some are stories about space-traveling agents who have psychic abilities as part of their job skills.
But at some point psychic abilities in a story stopped automatically tagging the story as "maybe" science fiction or fantasy and instead immediately moved the story into the realm of fantasy. I heard one author mention that her space opera (with space ships and numerous alien races) was labeled "space fantasy" because some of her characters had powerful ESP.
There are some times, though, when the author does have the say as to what the story will be labeled. Sarah Beth Durst's newest book, Chasing Power has one character with telekinetic abilities and another with the ability to teleport. While I was reading it, I was thinking it was science fiction and then the abilities were described in the book as "magic." The rest of the excellent book has adventure and even more magic, so the fantasy genre is a good fit.
What do you think when you encounter ESP in a story? Are psychic abilities magic or science? Or do you depend on the other story elements before you label it? What are some of your favorites?