Common Telerin is the ancient tongue of the Teleri or Lindar clan of the Elves. It is the common ancestor of both Valinorean Telerin and the vernacular Elvish languages of the Westlands (the Sindarin dialects, and some of the Silvan Elvish like Nandorin).
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'.
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 In his personal early unpublished writings like The Etymologies, the only languages which possess the qualities of the Telerin languages are Old Noldorin and Noldorin.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)