Telepathy is the communication of messages or thoughts directly from one mind to another.
Generally, the messages are received and processed by the brain in a manner similar to ordinary sensory input, so in essence a telepath "speaks" and "hears" perceptually. However, the ordinary vocal and auditory mechanisms are not employed. Telepathy exists across space; it has appeared in various forms among various races.
Telepathy makes possible some forms of interaction that would otherwise be impossible. The Medusans are an incorporeal race so alien that the sight of them drives humans insane. Interaction with them was very limited, until 2268, when Miranda Jones established a telepathic link with Ambassador Kollos, proving that it could be done.
Extraordinarily powerful telepathic species exist. They are rare, but are extremely formidable, as they can immerse other minds in a realty of their own fabrication, much like a holodeck, but where the telepath makes all the rules. The xenophobic Melkot used their telepathic powers to punish aliens who encroached upon their space without permission (which, prior to 2268, they never extended).
Under rare circumstances, individuals can gain telepathic powers. In 2264, exposure to the galactic barrier altered Gary Mitchell, who developed telepathic powers. In 2266, Captain Kirk and the Enterprise took aboard a passenger, Charles Evans. Subsequent events revealed that Evans had been granted various psionic skills by the enigmatic Thasians, so that he might survive on their world.
Telepathic races such as the Betazoids, Vulcans and Ulllians usually develop a moral code (sometimes making it actual law) that precisely dictates under what circumstances the powers may be used. Telepathy, misused, can be the ultimate invasion of privacy, and for that reason, it is feared by many.
A few civilizations have developed mechanical telepathy. Landru, a gifted engineer and philosopher, constructed a computer able to telepathically control most of the population of the planet Beta III. The engimatic Flint built a series of robots that could be controlled telepathically, and that could respond to mental emanations consistent with distress or danger. Sargon, Thalassa, and Henoch, the last survivors of an extremely advanced civilization, knew how to build robots their minds could inhabit and control. Furthermore, the inhabitants of the Think Tank met by Voyager had built a mechanical means for communication, as their species differed so profoundly that they never would be able to communicate by verbal means.
A touch telepath can communicate directly with the mind of another being, but only if in physical contact. The Vulcan mind meld is a form of touch telepathy (although there is more to it than that).
Some telepaths can send and recieve feelings and emotions. Sometimes this is part of a telepathic skill, and sometimes it is their only skill. In the latter case, individuals are referred to as empaths. Examples of empathic species include the Lumerians, Napeans and Deltans. Human-Betazoid hybrids tend to be only capable of empathic abilities with regard to non-Betazoids.
Some telepaths can exchange thoughts only with others of their own species (Cairn, Lumerian), some can send but not receive, and some can receive but not send. Betazoids, for one example, can send their thoughts only to other empaths and telepaths, but can receive thoughts and feelings from almost any being's mind, psionically gifted or not.
In some races, telepathy is present in all healthy individuals (those not born with congenital problems and not the victims of accidents or disease). In other species, such as Humans, telepathy appears only infrequently. And certain species, notably the Ferengi, are never telepathic. Often such species cannot be read even by telepaths normally capable of reading alien minds.
Telepathy is possible because the sentient mind itself is independent of the brain. It exist as a "psionic field" in space and the brain functions by simply "tuning in" to this field like a radio tuning into the energy signal from a station, the DNA serving in a similar capacity. The body's bio-energy field acts as the "antenna", a bridge to the "mind field" that stores all aspects of the consciousness. This being the case, each mind has its own separate field. While existing in the same space, each field is effectively separated from other minds, by virtue of having a different frequency - one unique to itself.
The mind fields exist in an alternate dimensional plane. From such a dimension all points in this realm are accessible from one location in it and vice-versa. In effect, the psionic field can be said to be all-pervasive relative to this universe.
This other dimension exists outside of space-time and is therefore not an actual physical construct. People have their own individual characteristics, but are "composed" of the same "substance" as the rest of the alternate dimension. In other words, their own "personal" storage space, rather than being "partitioned off" from the rest of the alternate dimension, is actually part-and-parcel of the "fabric" of the parallel realm. They are able to access their thoughts, memories and personality strictly by virtue of their physical/energetic resonant signature, which is matched to their own field.
Telepathy is analogous to cordless phones/cell phones and "crossed signals". While everyone's physical/energetic signature is unique, "overlapping" is possible, which allows someone to "tap into" the psionic field of someone else, providing that he or she can "get into resonance", or "tune" to that person's bio-energy field signature.
During telepathy the psionic field serves as the "medium" through which unspoken thoughts and feelings are communicated through space. Telepathic humanoids can tap into another field through a kind of sense organ located in the brain (e.g. the paracortex). In the same manner that human eyes can sense portions of the electromagnetic field, telepaths can sense different portions of the psionic field.
A telepath's neurophysiology enables them to develop a "resonance" between themself and another person, so their resonant signatures "overlap" to a certain extent which "re-tunes" their mind frequencies to match, allowing for the "sharing" of thoughts, emotions and memories, and, in some cases, even their personality traits, etc. such as in a Vulcan mind meld.
Incidentally, this also explains the latent capacity for genetic memory in humans: as the DNA is so instrumental in the "tuning" process, people of close genetic relation, thus sharing elements of their DNA, may "tap into" ancestors' memories. Even if an ancestor is dead, their memories still exist in the parallel realm.