Telepathy- Communications with the mind. Speaking, reading minds, altering memories, or reading emotions.
Telekinesis- Moving objects. At the high end, teleportation.
ESP- knowledge of things through unknown means. Battle sense (predicting opponents moves), and Akashic (learning new information by meditating on it) are included in this.
Ki- manipulating the energy of life. Healing, increasing physical attributes, sensing life energy, and the manipulation of energy outside the body are all in this category.
Elemental- Rare gift to alter a specific thing, sometimes thought of as a form of TK or Ki powers. That thing could be anything from water to light to radiation, and is usually developed as a response to that thing threatening your life.

Recorded via hologram in the cave:

((The Sky-cloud people came to this land, fleeing those who wish to kill them. Their chiefs had magic, and used it to control the people. Those who would not obey were killed, so the people lied and lived. In time they grew tired of lieing, so they fought. They lost, but stole sky boats to come here. They built the Doorway to move them deep beneath the ground, and there they hid until they had gathered their allies to join them in the fight.))

Computer records:

A group of mammalian aliens (a type we don't have, with cat-like eye, pig-like skin, and horns) came from their home system (which lies within a nebula) to this system to flee the oppressive Nobles of their homeworld. Throughout their recorded history, those with psychic power became nobles. Those that didn't have power were commoners, and treated as little more than slaves. Should a commoner be found who was an Indomitable Mind, they executed them, fearing that they would turn against the Nobles. The people learned to hide this fact and, in time, the number of people who were Indomitable Minds greatly increased. Around the year 900 AD a group of IMs decided that they were tired of being treated as slaves and started a revolt. The nobles, with their psychic powers, slaughtered most of those that stood up against them.
The IMs knew they would be defeated if they continued down this path, so they stole a starship and fled their homeworld. Continuing to raise an army via FTL communications with their people, they came to Earth. Afraid that the Nobles would follow them, they set up two teleporters, one in a cave, and one in a giant underground cavern. To protect the one in the cave from anyone who wasn't a IM they placed a device beside it that places the thought of leaving in your mind and tries to make you obey that thought. They then moved under ground in order to try and learn the secret of psychic power.
A few months after they arrived a group of Native hunters entered the cave to stay out of a storm. While many of them felt like they should brave the storm and leave, the leader of the group calmed them. As they sat near the fire, he went deeper into the cave, where he found a device made out of materials he'd never seen before. When he stepped inside of it, he was sent to the cave. At first the aliens were frightened, but soon one of them came upon the idea of studying the Native. Within him they found dormant Psychic power and, after studying him, they were able to unlock those powers with their technology.
In time they learned to give themselves powers as well, though, as they had no gift for it, their powers were weak. They sent their method to their people on the homeworld and, when the rebels there informed them that they had their own skilled psychic warriors, the alien left for their homeworld.
The base became a sacred place for the Natives in the area, and a way to test young warriors, but those beliefs all but disappeared in modern times. When we went looking for a base, though, we were lead there by an IM Native American who remembered the stories of his ancestors.