This article describes a concept which is mentioned in J.R.R. Tolkien's works, but was never given a definite name.
The stage of "Old Sindarin" refers to a period where the Sindarin words had not undergone lenitions of their consonants, nor affections of vowels by i or a, nor apocope of final vowels. The first changes that already occurred in Old Sindarin, which made it differentiate from Common Telerin, was the aspiration in some sounds; this can be seen in Telerin alpa but OS alpha > S. alph. Further changes followed in the process of time.
This language was spoken in Middle-earth during or after the Great March, perhaps by the Eglath. However when the Noldor returned to Beleriand at the rising of the Moon, the language had already fully evolved into Sindarin.
 Other versions
In published writing referring to the early legendarium, only the term Old Noldorin was used. In that early conception, it was the language spoken by the Gnomes in Kôr, (in contrast to the later legendarium of The Silmarillion, where the Noldor spoke Quenya both in Valinor and later in Middle-earth). Old Noldorin does show some similarities to Quenya and Telerin.
Most of the corpus of ON is given in The Etymologies and their purpose is to illustrate the etymology of some Noldorin words. Even after Tolkien renamed "Noldorin" into "Sindarin", almost all rules of etymology and derivation have been retained the same (see "Noldorin" for the discussion of some minor differences).
More recent works like Quendi and Eldar show ancient forms of Sindarin that seem to possess the same structure and to belong to the same evolution as the Old Noldorin of the Etymologies. It is therefore possible that an ancient form of Sindarin similar to what Tolkien imagined as "Old Noldorin" did exist in the First Age of his later concepts; but Tolkien did not refer to that stage as a stable, identifiable language and did not go to the process to name it; making "Old Sindarin" to seem a non-canon, but plausible name for it.
Note that in these recent sources, Tolkien provides these forms with an asterisk as "unattested", suggesting that they belong to an unwritten stage of the language, before the invention of the Cirth. This is not the case in the "Old Noldorin" forms of the Etymologies where they are unasterisked, supposedly written with Tengwar in Kôr.