Konor were a part of the Samdian race that developed unusual telepathic abilities by the mid 24th century. Konor is a Samdian term for "those with souls." As they originated from the planet Jokarn, they maintained golden-green skin and lavender hair all throughout their lives, unlike the other Samdians on the other planets in the Samdian sector. Also unlike other Samdians, they were more slender and willowy, and aside from sandals, they wore nothing at all (unless they were in combat) and did not adorn themselves or grow beards. They had no need for individual names, claiming they identify one another by the essence of their souls. They were governed by the Konor Council of Elders.
When the Konor members started to develop their unique mental broadcasting, they were naturally drawn to those they could communicate with. As the group grew in size they came to believe that Providence - the Samdian God - had done this to signify that they alone possessed souls, and that those who could not transmit their thoughts as they did had no souls, and were thus Ikonor, or those without souls. This made them believe that things made by others were meant for them, and Ikonor were to be their servants. The Konor then took to killing or enslaving those who did not share their telepathic abilities, took members of other Samdians who had their abilities, and waited for Konor of other species to contact them.
While the Federation was unable to intervene militarily due to the Prime Directive, Lieutenant Commander Data discovered that he was also able to receive the thoughts transmitted by Konor while being totally unable to perceive transmissions from traditional telepaths - such as Vulcans or Betazoids. With this knowledge he and Lieutenant Geordi La Forge were able to build a transmitter which Data was able to use to communicate with the Konor.
In an alternate timeline created when Data was transformed into a human by the Elysian gods, Data and the crew of the Enterprise were unable to resolve the crisis. Unable to find a peaceful solution the Enterprise left the Samdian sector.
Eventually the Konor conquered the remainder of the Samdian sector, using weapons they purchased from the Ferengi to do so. The Konor pretended to seek another deal with the Ferengi, but then captured the Ferengi ship sent to negotiate the deal with them - gaining shields, warp drive, and phasers in the process. The Ferengi attempted to retaliate but ran afoul of Romulan territory in doing so. The situation rapidly spun out of control and flared up into a full scale war that devastated most of the Alpha Quadrant cultures.
Fortunately this timeline was prevented from occurring when the Elysian gods returned Data to his android form and sent him back in time fifty days to the point before he discovered them. (TNG novel: Metamorphosis)