(covers information from several alternate timelines)
This page contains information regarding Star Trek: Discovery, and thus may contain spoilers.
The Betelgeusians were an ornithoid species, who by the mid-2270s were diplomatically associated with the United Federation of Planets and had members serving in Starfleet. The Betelgeusians spoke the Betelgeusian language.
A Betelgeusian named Valentine was aboard the USS Hiawatha when it crash landed on an interstellar asteroid in 2256. Commander Jett Reno managed to keep him alive by hooking up his heart to that of a dead Bolian until their rescue by the USS Discovery in 2257. (DIS: "Brother")
In the alternate reality, Lieutenant Commander Yor was a Starfleet operations division time soldier in 2379. He traveled from that reality to the prime reality while also traveling forward in time. (DIS: "Terra Firma, Part 1")
- Star Trek: The Motion Picture
The Betelgeusians were designed by Fred Phillips and Robert Fletcher for The Motion Picture. With the approval of Gene Roddenberry, the Betelgeusians, as well as numerous other new species in the film, were named by Fletcher and provided a backstory by him.
The name "Betelgeusian" was introduced to the world nearly a full six months before the release of The Motion Picture. During her visit to the set of The Motion Picture, Susan Sackett noted many "curious creatures", including the "bird-like Betelgeusians." (Starlog #23, June 1979, p. 31.)
In an interview with Robert Fletcher – published as part one of "The Star Trek Costumes", in the February 1980 edition of Fantastic Films – the Betelgeusians were described thus; "Although humanoid in conformation, they are evolutionary derivatives of a race of creatures combining characteristics of eagles and leopards." 
In addition to the above, a broader description was later revealed by Fletcher, stating "Name from the real star. All tall, 7 feet and over. [...] Derived in evolution from giant leopard-like birds, have claw and bone structure like condor but walk upright like leopards." (Starlog #33, April 1980, p. 51; The Making of Star Trek: The Motion Picture, p. 130; Star Trek: The Magazine Volume 2, Issue 9, January 2002, "Who is that Alien?")
Two Betelgeusian-related lots were included in the It's A Wrap! sale and auction, each containing an original Betelgeusian mask. The first lot sold for US$202.50.  The second – which contained three masks, original jewelry, and a bronze-colored robe – sold for US$732. 
Valentine's species is not mentioned in DIS: "Brothers", and although StarTrek.com described him as a Lurian, his make up is identical to Cosmo Traitt's in DIS: "That Hope Is You, Part 1". Prop Master Mario Moreira identified Traitt and Traitt's weapon as Betelgeusian in TRR: "That Hope Is You, Part 1".
According to the Star Trek: The Next Generation - Player's Guide, Betelgeusans come from the planet Betelgeuse IV, which they call "Hav'a'halar". Betelgeusans do not make any distinction between history and mythology. Every Betelgeusan chooses one of five heroes from their mythology to emulate: Altaa, Slichez, Lahile, F'ter and Doban. This practice is called j'tonka. Criminals follow a sixth evil hero named Cordban, while pretending to follow one of the others. Betelgeusan society is divided into social groups called naccords that emulate the same hero.
According to the novel Ex Machina, Betelgeusians have two mouths, an eating mouth and a speaking mouth. The novel The Face of the Unknown reveals that Betelgeusians actually originate from a main-sequence star system close to Betelgeuse. They migrated away from their homeworld centuries ago, since they were aware that their home would be destroyed when Betegeuse goes supernova. Betelgeusians are now a nomadic species that live on spaceships and space stations called argosies.