The Sindar, known as the 'Grey-elves', are a group of the Teleri that turned aside during the great journey to Aman. They were the followers of Elwë (Elu Thingol) and they stayed behind when their lord went missing. The Sindar also comprise the Falathrim, a sub-set of elves that were persuaded to stay behind by the Maia Ossë.
The center of Sindarin culture was the kingdom of Doriath from which Thingol and his wife Melian the Maia ruled. Thingol and the Sindarin elves were suspicious of the Noldor's return to Middle-earth and when the truth of their return was learned the Noldorin tongue of Quenya was banned from the kingdom of Doriath; only Sindarin was allowed to be spoken.
At the end of the First Age and the destruction of the lands of Beleriand, the Sindar settled in a small area that was a remnant of Beleriand, the land of Lindon, and some traveled east and settled around the Hithaeglir (Misty Mountains).
During the Second and Third Ages, the 'Sea-longing' was awoken in many of the Sindar and they sailed west to Aman from the haven of Mithlond in Lindon which was held by Círdan. Celeborn is noted as one of the last Sindarin lords to leave Middle-earth, which happened early in the Fourth Age.
The Sindar are not as gifted as the Noldor in lore and craftsmanship but they are extremely gifted in music, and the elves that followed Círdan, the Falathrim, are the greatest sailors and ship builders among their race.