Mind-Speaking is the most basic form of telepathy, and is the one people most commonly associate with the term. You sit there, I sit here, and we talk to each other without moving our lips. Such conversations are typically written like ordinary dialogue, usually in italics.
Dude. I’m trapped in this prison cell, can you still hear me?
Yeah, dude, I’m in the one right next to you.
Mind-speaking may or may not be achievable over long distances, depending on the writer, the world, and the rules of the world. Commonly used when communicating with dragons. Examples of books that use mind speaking are Eragon, and The DragonKeeper Chronicles.
Mind-Reading is the next step up, the one usually found in more science fiction type works. There are two forms of mind-reading; the Touch Telepath, and the Intrusive Psychic.
The Touch Telepath requires physical touch in order to communicate with another’s mind, and generally must obtain permission of the person who’s mental privacy they’re invading. There are certain safeguards allowing you to sheild some thoughts from the probing of the mind-reader, such as imagining a closed door. Examples of Touch Telepaths are the Doctor from Doctor Who, Spock from Star Trek: The Original Series, and (in this case a villain) the Wraith from Stargate: Atlantis.
The Intrusive Psychic has advanced mental abilities, and does not usually require any sort of physical contact to invade their victim’s mind. The victim may or may not be aware that their mind is being read. The Intrusive Psychic rarely asks permission, and often uses their skills to impress others. Examples of media featuring Intrusive Psychics are Inception, and some episodes of SeaQuest; DSV.
Mind-Control is one form of telepathy that is universally acknowledged as wrong. Mind-Control is the result of a very powerful mind-reader, often with the aid of advanced technology or magic, taking over the mind of their victim, so that the victim is forced to do whatever the mind-reader tells them. Also a form of hypnosis. They may subtly alter how the mind functions so that the victim will act of their own free will in the interests of the mind-reader.
Memory-Sharing is an extremely powerful means of communication between two telepaths. The memories shared directly can be very strong, sometimes causing identity confusion. The only examples of Memory-Sharing that I’ve encountered is the fictional series The DarkTrench Saga
Group Memory is a phenomena observed only in telepathic races and cultures. While not yet forming a group mind each individual will have an awareness of their people as a whole, often being able to access memories seen by their ancestors. When a great devestation occurs most members of the race can usually sense it. Examples of telepathic races known to show these aspects are Vulcans (Star Trek), Time Lords (Doctor Who), and Na’vi (Avatar).
Group Minds are a more sinister aspect of being part of a telepathic culture. It is the biological equivalent of linking several ordinary computers together into one, huge supercomputer. Individuals loose a large part of their identity and, in some cases, may loose any sense of individualism at all. The minds function as one unit, with each member acting for the good of all. Examples of group minds are the Borg (Star Trek), the Cybermen (Doctor Who), and the Attic (Dollhouse – maybe be incorrectly categorized).
Other Aspects of Telepathy
Other aspects of telepathy are used in fiction, although, strictly speaking, they’re not telepathy as such, but more of general psychic ability.
The telepath who can manipulate matter into other shapes may or may not actually be channeling energy from another source. Also known as transmutation, this skill is fairly rare. It may also include the ability to cast an illusion.
Usually seen as a strictly technical or magical ability, teleportation can be achieved in some instances by telepaths. Usually by simply wishing themselves elsewhere they can succeed in changing their actual location. This is about as plausible as “lifting yourself by the seat of your pants” but by employing other senses such as television, super hearing, and telesmelling a strong enough telepath can move himself and anyone else who happens to be coming along to another location in a method very similar to many teleportation devices with much less trouble and expense. Examples: Harpist in the Wind (Riddle of the Stars).
Networking with Electronics
Often an aspect of being a Cyborg, this may or may not actually qualify as telepathy. The individual who can directly connect to a computer usually does so by means of a neural implant, rather than by more traditional telepathic methods. Yet it could be said that such implants are themselves a means of enabling lesser mind to achieve an aspect of telepathy they might not otherwise possess. Examples include R2D2 (Star Wars), and the DarkTrench Saga.
Usually an instinctive ability the individual is born with, shapeshifting can often be obtained by mental discipline. Often adopted by energy beings, it is sometimes seen as the ultimate victory of mind over matter. Examples of shapeshifting (or appearance altering): Riddle of the Stars, and Star Trek. Examples of energy beings: Stargate: SG-1, and The Adventures of Lucky Starr (Isaac Asimov).