The Sheliak were an intelligent non-humanoid classification R-3 lifeform whose homeworld was located in the Shelia star system. They were warp-capable and their governing body was called the Sheliak Corporate.
Similar to dealings with other non-humanoid races, the Federation struggled to deal with the Sheliak, due to a lack of a common point of reference. The Sheliak considered the Federation's languages irrational, and while they had learned several Federation languages, the Sheliak language eluded the Federation. Even telepaths had tried to communicate with the Sheliak without success. Furthermore, the Sheliak considered Humans lower lifeforms and held no compunctions about eradicating a Human "infestation".
Despite these difficulties, the massive Treaty of Armens was signed between the two powers. The Sheliak insisted on a level of complexity in the treaty which would eliminate ambiguity, as they considered Federation language to be "irrational". Three hundred and seventy-two Federation legal experts were sent to draft the treaty. In the treaty, the United Federation of Planets ceded several class H worlds to the Sheliak.
In 2366, the Sheliak decided to colonize one of their planets, Tau Cygna V. Finding a Human colony there, they sent a message to the Federation to remove them, their first attempt to establish contact in one hundred and eleven years. The USS Enterprise-D received the message and confirmed the presence of the colony. The Sheliak gave them only a few days to relocate the colonists, but logistically that was impossible for the Enterprise. The Sheliak showed no concern for this, and initially refused to grant the Federation any additional time, eventually relenting after Captain Jean-Luc Picard asked for third-party arbitration in the treaty, and used it to force the Sheliak to cooperate. The Sheliak colonists referred to themselves as "the Membership". The Sheliak colony ship turned back from their journey to the planet to return after three weeks. (TNG: "The Ensigns of Command")
According to an internal reference document, listing all the aliens seen in Star Trek: The Next Generation, which was circulated around the time of the production of TNG Season 5, the Sheliak were given the description of "a technological, non-humanoid lifeform of classification R-3, in appearance a wrap of black, mucous-secreting folds appearing as at least a head."
Tau Cygna V was saturated with hyperonic radiation, making it a class H world and one of the worlds ceded in the Treaty of Armens. Although it is possible that the Sheliak intended to terraform the planet for their needs, the episode suggests that hyperonic radiation was of no concern to their biology.
While Data specified the term "Sheliak" to be plural, the Treaty of Armens also at one point referred to them in plural as "Sheliacs".
In a blooper from "Starship Mine", Brent Spiner, as Data, stated, "There is only one way to measure the body temperature of the Sheliak and, uh, I can't say it in mixed company." ("Gag Reel", TNG Season 6 Blu-ray special features)