Sindar (meaning "Grey People", singular Sinda, although the later term was not generally used by Tolkien) were Elves of Telerin descent. They are also known as the Grey Elves. Their language is Sindarin. The Kings of Doriath, Elu Thingol and his grandson Dior Eluchíl, were the High Kings of the Sindar.
The Sindar are "grey" perhaps because they were not Elves of the Light, never having actually set foot in Valinor, but neither were they Avari, since they did accept the invitation and set out on the journey (and therefore were technically counted among the Eldar). They are sometimes referred to as "Elves of the Twilight". The term "grey" may also refer to Elu Thingol's grey cloak. The term Sindar is actually not Sindarin in origin: it is the Quenya name devised by the Noldorin exiles. It may also be related to Lindar, the Telerin name for themselves. The name that the Sindar used for themselves was simply Edhil ("Elves", singular Edhel).
The Teleri were the greatest host of the Eldar. They had two kings, the brothers Elwë (known as Elu Thingol in the Sindarin tongue) and Olwë. When the Teleri reached Beleriand during the Great Journey from Cuiviénen, Thingol went wandering in the forests as was his wont. In the forest of Nan Elmoth he met Melian, one of the Maiar. They fell in love, and with Melian, Thingol stood spellbound in Nan Elmoth for several years.
In the meantime, Olwë and many of the Teleri could not delay longer, and went to Aman without Elwë and his following. Elwë's followers stayed in Beleriand, to search for their king. They later became known as the Eglath (the "Forsaken"). At long last he awoke from the spell and set up a kingdom in the midst of Beleriand: Eglador ("Land of the Forsaken" or "Land of the Elves", the etymology is not clear). The Dwarves of Nogrod and Belegost in the Blue Mountains were contracted to aid in the building of the city of Menegroth.
Other Teleri also stayed behind: these were the friends of Ossë the Maia, who had fallen in love with the shores of Middle-earth, and did not wish to depart. Their leader was Círdan, and they established cities at Eglarest and Brithombar. They were known as the Falathrim, or "Elves of the Falas". They were not part of the realm of Eglador, but still took Thingol as their High King.
The Teleri of Eglador, the northlands, and the Falas were collectively known as the Sindar in later days, because they developed a civilisation all its own, which almost equalled that of the Calaquendi or Light Elves of Valinor.
A last group of Teleri in Beleriand were the Laiquendi or "Green Elves". They were descended from the Nandor, which had split from the Great Journey before the Misty Mountains, and gone south along the Great River. A part of them, under Denethor son of Lenwë, crossed the Blue Mountains eventually, and settled in Ossiriand, or as it was later known Lindon ("Land of the Singers"). They remained a people apart for long, although many of them removed to Thingol's realm after Denethor was killed.
Just before the arrival of the Noldorin exiles, the Dark Lord Morgoth returned to his old stronghold of Angband, and his activities increased. Thingol had Melian use her magic to create a girdle of bewilderment around Eglador, so that nobody could enter without the king's permission. Ever after it was known as Doriath ("Land of the Fence"). Thingol remained High King of the Sindar and nominal Overlord of Beleriand, although especially the Noldor following of the sons of Fëanor usually ignored his commands.
The language of the Sindar diverged from common Telerin over the long ages they were sundered from their kin, and became known as Sindarin. By the time the Noldor arrived in Beleriand, the languages had become mutually unintelligible, but the Noldor were quick to learn it. In the Second and Third Age, Sindarin became known as the "Noble Tongue", and became the Elvish tongue used in daily speech throughout Middle-earth (helped by the decree of Thingol, who forbade the use of Quenya in his realm). It was also adopted for daily use by the Númenóreans, and remained somewhat in use in the realms-in-exile of Gondor and Arnor.
Some of the few Sindar left during the Third Age were Thranduil, and possibly (though not probably) Celeborn. While the subjects of Thranduil were Silvan, and those of Celeborn Nandorin, the Sindar were apparently recognized as the ruling class by the lesser green-elves.