This page contains information regarding Star Trek: Discovery, and thus may contain spoilers.
In 2151, Hoshi Sato noted that the frequency distortions of the symbiotic lifeform found aboard Enterprise NX-01 looked a lot like the phonetic patterns in certain Andorian dialects. (ENT: "Vox Sola")
In 2154, Ambassador Soval told Shran that his name would be used in the Andorian language as the word for fool in the future just as "Nirak", a Vulcan who failed to stop an invasion, meant the same word in the Vulcan language. (ENT: "Kir'Shara")
In 2257, when the universal translator on the USS Discovery malfunctioned, one of the many languages the system translated verbal and written displays into from Federation Standard was the Andorian language. (DIS: "An Obol for Charon")
In 2287, while in the observation room of the USS Enterprise-A, Sybok explained to Captain James T. Kirk that there were many different names for the location from which creation emerged in different cultures. As Sybok explained, the name for this place in the Andorian language was unpronounceable. (Star Trek V: The Final Frontier)
According to the non-canon DS9 Star Trek: Section 31 novel Abyss, the novel This Gray Spirit and the "Andor: Paradigm" novella in Worlds of Star Trek: Deep Space Nine, Volume One, the primary spoken language of the Andorians was called "Andorii." An Andorian language glossary is available at the back of the latter novel. The current novel-universe also uses a different naming convention for Andorian characters.
The Star Trek RPG supplement Among the Clans from Last Unicorn Games, on the other hand, claims the primary language was called "Graalek"; this looks very different from the written language seen on screen, consisting of curved horizontal lines and circles, though this can be explained as two different languages used on the planet. This alphabet appears on some of the banners at the Interspecies Medical Exchange conference in ENT: "Stigma". Among the Clans attempted to explain the Andor/Andoria inconsistency by claiming that in Andorian, "Andor" merely means "World", but adding "-IA" (Andoria) promotes a word to prominence over other words.