Sindar is the name of the people of King Elu Thingol of Doriath. They were Telerin elves who stayed in Beleriand rather than go with their brethren over the sea to Aman. Called 'grey-elves', they were accounted part of the Eldar (having started on the Great Journey from Cuivienen) but also part of the Moriquendi, because they (other than their king Elwë) did not see the Light of the Two Trees of Valinor. Despite this, they became a powerful people. They covered the center lands of Beleriand and became mighty under the rule of Thingol and the tutelage of Melian the Maia their queen. They also traded extensively with the Dwarves from Nogrod and Belegost, and developed large armories and stores of weapons. By the time of the Exile of the Noldor (the deep-elves) the Sindar were mightly indeed.
Kingdom of the Sindar
The Sindar fought the First Battle of Beleriand against the forces of Morgoth and the legions of Angband. They were victorious, but they suffered such loss that Melian created a great spell that encircled the realm of Thingol, called Doriath. This enchantment protected the land and was called the Girdle of Melian. No person could pass the Girdle save with the permission of Thingol or Melian, or if some greater fate drove them (Beren Erchamion was one such.) The destruction of Doriath was only wrought by the pride of Thingol and his keeping of the Silmaril. Upon Thingol's death at the hands of the jealous Dwarves (the full story of which is in The Silmarillion), Melian returned to Valinor and the Girdle of Melian was diminished. The Sons of Fëanor then assualted Doriath and slew Dior, Thingol's heir, in an attempt to gain the Silmaril.
In The Lord of the Rings the "Elven" tongue spoken is the Sindarin language (even among the remaining Noldor.) This was because of a decree by Thingol in the First Age. Upon being told of the Kinslaying at Alqualondë Thingol decreed that he would never have friendship with the Sons of Fëanor and that the Noldorin tongue was never to be spoken in his hearing. Although Noldorin remained as a language of lore among the lords of that people, Sindarin quickly became the dominant Elven language. It is this language which was spoken during the Third Age, in Lothlorien, Imladris, and by Cirdan of the Grey Havens.
After the War of Wrath and the defeat of Morgoth, most of the remnants of Doriath went over the Sea and dwelt in Aman. Those who remained stayed with Galadriel and Celeborn or with Elrond, both of whom were of their kin. In the Third Age after the defeat of Sauron most Sindar lived in Imladris with Elrond or at the Grey Havens with Cirdan. By the end of the Third Age and the departure of Cirdan and Elrond few, if any, Sindar remained in Middle-earth.
Sources
J.R.R. Tolkien.
The Silmarillion.
New York: Houghton Mifflin, 2001
J.R.R. Tolkien.
The Lord of the Rings.
New York: Houghton Mifflin, 2001