The elves are one of the most impressive race of beings in Middle Earth, with a long and complicated history. They, like the realms of Men from Gondor, Rohan etc, have their own different factions within the larger group, with the four most known being Vanya, Noldor, Telari and Silvan, but there are many more. The most common characteristic known about the elves is that they are immortal, and so, have been around for thousands of years, some of them since the First Age of Middle Earth. But of all these mighty beings, who is the oldest?
When referring to the elves who have lived in Middle Earth specifically, the answer is most likely Cirdan the Shipwright. Consulting The Silmarillion about other elves outside of Middle Earth, including in the Undying Lands, and in Numenor before the fall will give a different answer, as there are many ancient elves in Tolkien’s legendarium, but Cirdan is believed to be one of the original 144 elves who were awoken by Eru Iluvatar near Cuivienen, and are the founding fathers from which the entire race of all elves descended, although whether or not he was among this legendary few has been debated. It is confirmed by Tolkien himself though, that he is kin to Elwe and Olwe, who were the kings of the Teleri, the third clan of the original 144 to make the journey from Cuivienen to Valinor.
Cirdan is probably most known within the Lord of the Rings trilogy as being the person that gave Gandalf Narya, the elven ring of fire, thus strengthening Gandalf’s light magic, which saves both Thorin and his company, and the fellowship taking the ring of power to Mordor on many many occasions. The bestowing of the elven rings by Celebrimbor was a high honor, and these precious items were only given to the utmost trusted among the elves, Cirdan, Gil-galad, and Galadriel, 3 of the oldest and wisest of their people in Middle Earth.
Celebrimbor chose these 3 in the knowledge that they would keep the rings protected, hide them from Sauron when he was seeking control with the master ring, and then use them for good when it was safe to do so, after Sauron had fallen from power. Knowing Gandalf’s good heart, and his duty as a maiar to preserve and guide the mortals of Middle Earth, Cirdan then passed on the ring to the gray wizard, knowing that it would suit the magicians' purpose better than his own.
Cirdan known as a Shipwright for 2 reasons. The first of which is that he was sent great visions of ships by the Valar, giving him his purpose, and telling him to build ships that would be strong enough to sail the unchartered seas. He had the elven gift of foresight, but these visions helped him to prepare to save the world many times during the Wars of the Noldor. He became a great sailor, and eventually made his way to Middle Earth.
Although his heart always longed to go back to the Undying Lands, he knew it was his duty to stay and protect those who needed him. Thus, the second reason he became known as the greatest shipwright was because of his position as the Lord of the Harbor. As the guardian of the lands of Valinor, he was in charge of aiding the elves in their departure west. He helped all of the wounded and troubled immortals take their final journey across the sea, and he was eventually rewarded for his years of service by finally being allowed to join them, sailing across with Celeborn to reunite with Galadriel at the close of the Third Age. He can actually be seen in Peter Jackson's film adaptations of the Lord of the Rings, because he is the old elf stood in the background when Frodo and Bilbo take the ship west.
Cirdan is one of very few elves to reside in Middle Earth long enough to reach his third cycle. The first cycle is youth, the second is adulthood, and the third is old age. He was even old enough to grow a beard, which is very rare and unusual among elves. He is thought to be over 10,000 years old by the time the War of the Ring takes place, and in all those many long years, he is credited with saving the entire elven race, selflessly putting his own wants and desires aside for the good of all, and of being a vessel through which the divine could communicate with the rest of the beings in the world. When he finally leaves, he takes with him the only remaining memory of the Elder Days, and Middle Earth passes into its Fourth Age with only the stories to remind them of the honor and majesty of days passed.