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Psychic power
"Also? I can kill you with my brain."
— River Tam, Firefly

Telepathy, clairvoyance, pyrokinesis — the powers are supernatural, but the names are scientific, which is good enough for soft Sci Fi. Pointy-eared elves mumbling ancient spells on shiny spaceships would be incongruous; pointy-eared aliens reading minds on shiny spaceships doesn't raise any eyebrows.
In general, the more powerful and dramatic the psychic, the softer the Sci Fi — psychic powers are one of the most common ways of including Magic By Any Other Name and recycling Magical Tropes in a scientific environment. The extreme cases are largely confined to the horror, Cosmic Horror Story and superhero genres (with exceptions, such as Star Wars), but the weakest powers can crop up even in mainstream shows.
In order of increasing power, the standard abilities are:
Clairvoyance/Clairaudience — also called TeleSense, Remote Viewing, Remote Sensing, Extra-Sensory Perception or ESP. Seeing (and sometimes hearing, or using other senses, including ones that aren't part of the standard package) far-away places, localizing specific persons one concentrates on, usually involves a trance state; the amount of control over what is seen can vary wildly, depending on the talent and training of the psychic and how the power works in your 'verse.
This can get broke down even further into Clairaudience, the psychic ability to hear things that are inaudible, Clairempathy the ability to feel other’s emotions to an abnormal degree, Claircognizance, having knowledge one has no normal way of knowing, Clairsentience refers to a psychic’s ability to pick up sensations and relate messages from those sensations, Clairgustance, which is psychic supertaste, and Clairalienence, psychic supersmell. Clairvoyance is sometimes aided by hypnotic trances – a process called metagnomy – famously used by “real life” psychic Edgar Cayce.
Variants of this include Aura Vision — varying in power from just visible to reading someone like a book; can encompass a wide variety of other sensing powers, usually tied to Pineal Weirdness and the Third Eye. Astral Projection and BiLocation (the ability to create a double of oneself, physical or otherwise, in another place and/or time). The latter may overlap with some forms of teleportation.
In dream form, this is Dream Spying.
Precognition — seeing the future in prophetic visions, sometimes in allegorical pictures. Often leads to a Prophecy Twist or Self-Fulfilling Prophecy. Dreaming of Things to Come is a common form.
Retrocognition/Postcognition — seeing things that happened in the past— not including watching security footage, natch. Often happens at crime scenes and may be considered a subset of clairvoyance.
A more limited version of this is called Psychometry, or object reading, which "reads" the past of an object. Sometimes also includes aspects of Empathy, by picking up an imprint of strong emotions from the object left by the person who last handled said object. Can include sensing the "auras" of places, too, especially if something violent happened there recently.
Certain versions of Pyschometry have combat applications in that a Psychometer who picks up a weapon may theoretically be able to absorb the skills of anyone who's ever wielded that weapon.
Another fairly common variant is sensing the memories of one's ancestors or (in universes with reincarnation) past lives, similar to Genetic Memory.
This can manifest as Dreaming of Times Gone By.
Empathy — the talent of The Empath, the ability to sense another person's emotional state. At times, even disturbances in the force and how strong the force is with someone. Sometimes includes the ability to send emotions the other way.
Telepathy — mind reading, can sometimes also transmit thoughts or implant suggestions, etc. Telepathy is the psychic power most commonly attributed to aliens or "advanced" humans.
Mindspeech is the ability to have conversations psychically. It usually requires a common language. Sometimes, though, as it is supposed to be direct thought-to-thought contact, any and all language barriers are completely overcome.
A Psychic Link is an intense, (usually) permanent telepathic connections between two (usually) partners. Subtropes include Twin Telepathy, Mental Fusion, Mindlink Mates, and Bond Creatures.
A variant is the suggestor, who can control or at least influence minds, but not read them. Often these suggestions are said to only affect the weak-minded.
Some telepaths have the ability to enter the dreams of people, to implant pictures into the mind of sleepers, or merely talk, but can do this only when people are asleep.
Then there are those who are able to harvest the target's dreams, and use them for their own purposes.
Similarly, another breed of telepath is the Master of Illusion.
Telepathy can often be stopped by Psychic Static and Psychic Block Defense.
Telepaths are often shown to be able to detect the presence of others; sensing their thoughts or merely the presence of a mind in a sort of Psychic Radar.
Telekinesis/Psychokinesis — moving physical objects by pure willpower. Can range from atoms to paperclips to cars or, in extreme cases, whole planets. Specialized expressions of it:
Pyrokinesis — setting things on fire. In slightly harder sci-fi, this will explicitly reference making the molecules in an object more agitated until it bursts into flames (essentially, that's the way a microwave oven works). Sometimes the fire itself can be controlled, changing size or even becoming a particular shape.
Similarly, Cryokinesis — freezing things. Slowing molecular motion until the object stops exuding heat, or just until it freezes solid. Often combined with condensing water from the air to form ice in thick coatings or free-standing shapes.
Fulgurkinesis/Astrakinesis - the ability to create electrical discharges and lightning bolts, and/or to control the flow of electrons inside machines. Sometimes encompasses the control of magnetic fields, too, if the author had a passing grade in high school science. (This is sometimes confused with the word "electrokinesis" which, incidentally has nothing to do with psychic powers.) A variant of this is Machine Empathy or Technopathy — the ability to mentally link to and control computer systems.
Biopsychokinesis, or Bio-PK — the ability to influence living tissue on the cellular or molecular level. Used for psychic healing, regeneration, Psychic Surgery or as a darker power the ability to kill people with your brain, traditionally by stopping their heart, but can also cause a massive stroke, simply shut the brain down, prevent the lungs from working...
Teleportation; Also known as Apportation — the ability to transport objects or people from location A to B, including oneself. A bit like the transporter in Star Trek. Usually the psychic will either call things to them or has to touch them to send them away to someplace else. If they're really powerful, they can move objects from/to both target locations from afar remotely.
Mediumship — the ability to see, communicate with or channel spirits, sometimes with the side effect that they'll follow the medium around in attempts to resolve their Unfinished Business.
Past Life Regression: A process of hypnosis or meditative visualization for reliving or discovering previous experiences of one's previous lives.
The first five powers are purely internal. There's no evidence they're being used apart from the occasional Psychic Nosebleed (and of course, the stance). The remaining powers have much more obvious effects. However, all these powers have stronger versions, found generally at the softer end of sci-fi. That is, strong clairvoyance is as good as X-Ray Vision, or even a Crystal Ball. Strong telepathy allows for complete Mind Control. Strong telekinesis or apportation can become a means of saying You Will Not Evade Me, and so on. The ultimate manifestation of psychic power is the ability to just make your thoughts into reality. As generally portrayed, all of these powers display No Conservation of Energy, though there are exceptions, especially if the writer wants to stiffen up the story's science yet still include cool Psionics; if so, expect terms like "quantum uncertainty", "observer effect" and "particle entanglement" to be slung around with abandon.

Stories can have both Psychic Powers and Functional Magic, but they'll usually be treated as fundamentally different. This does not preclude them from being related or even the same thing; supernatural powers are still supernatural powers, no matter what they're referred to as.
Compare Ki Attacks, Enlightenment Superpowers, and Functional Magic for other genre's "special powers." See also Mind Over Manners and Brain Critical Mass. 90% of Your Brain is sometimes brought in as an explanation for them. See also Phony Psychic and Not So Phony Psychic.