It descended from Primitive Quendian, the original language of all Quendi, or Elves, but split off Common Eldarin early, at some time during the Great Journey, perhas because the Lindar were further off from both Minyar and Tatyar.
Since many Teleri teams abandoned the March, and stayed in Middle-earth, all the languages of Beleriand of the First Age, like Sindarin, and the various Nandorin languages in the East, are descendants of Common Telerin.
One major Telerin change was that where Primitive Quendian used 'KW', Telerin had evolved it into 'P'.
Another significant change was the simplification of palatal 'J' into the vowel 'I'.
In Telerin of Aman
Other versions of the legendarium
The model of Common Telerin belongs to a newer part of Tolkien's writings. In his personal early unpublished writings like The Etymologies, the only languages which possess the qualities of the Telerin languages are Telerin and Old Noldorin (and by extension, Noldorin).
Fans who believe that those early languages are not obsolete in the later continuity, interpret them as dialects of Sindarin and/or Telerin languages in their minds. In that case those "anomalous" words that don't fit Telerin, are "updated" in Neo-Sindarin (for example *pess and *pindor)