Sindarin was originally the native language of the Elves of Beleriand, but by the Third Age it was the language spoken by nearly all the Elves of western Middle Earth. As Tolkien described it in The Lord of the Rings Appendix F:
Of the Eldarin tongues two are found in this book: the High-elven or Quenya, and the Grey-elven or Sindarin ...
The Grey-elven was in origin akin to Quenya; for it was the language of those Eldar who, coming to the shores of Middle-earth, had not passed over the Sea but had lingered on the coasts in the country of Beleriand. There Thingol Greycloak of Doriath was their king, and in the long twilight their tongue had changed with the changefulness of mortal lands and had become far estranged from the speech of the Eldar from beyond the Sea.
The Exiles, dwelling among the more numerous Grey-elves, had adopted the Sindarin for daily use; and hence it was the tongue of all those Elves and Elf-lords that appear in this history. For these were all of Eldarin race, even where the folk that they ruled were of the lesser kindreds (LotR/1127-8).
Sindarin was also widely spoken among men, especially among the Dúnedain, the high men of the West. In lands like Gondor it was usual for men to take names from the Sindarin language rather than that of the common Westron tongue. This includes names like Boromir, Faramir and Denethor, but among men such names did not always adhere to the proper rules of Sindarin (an Elf would have used the name Borovir). For similar reasons, most place-names in western Middle Earth are in Sindarin, such as Minas Tirith or the Hithaeglir (Misty Mountains).
I sometimes refer to Sindarin as “Common Elven”, as opposed to the “High Elven” tongue Quenya, though Tolkien himself never used this term. It is the Elvish language appearing most frequently in Peter Jackson’s Lord of the Rings and Hobbit movies, and is of greatest interest to those who want to emulate the daily lives of the Elves of Middle Earth, despite the difficulties presented by its complex grammar and many sound mutations.