In various of Tolkien's works, namely the Silmarillion and the Lord of the Rings, Círdan is a peripheral character that nonetheless plays a critical behind the scenes role in many episodes.
Círdan was one of the oldest and wisest Elves in Middle Earth. He was probably as old as Ingwë, Finwë, Olwë and Elwë, although he was not as high in rank as them. He made the great journey eastwards, but due to the fact that he was a Teleri, he chose to say behind in Beleriand looking for the lost Olwë rather then cross the ocean to Aman. He settled on the coast, and worked as a shipwright (his name, "Círdan" is actually Sindarin for "shipwright") and as a political and military leader of the Sindar elves that lived along the coasts of Beleriand.
After the return of the Noldor elves to Beleriand to fight Morgoth, Círdan led the Elves of the coast in many battles against armies of orcs and other minions of Morgoth. However, no stories about specific accomplishments of Círdan are told in the Silmarillion. For one thing, Morgoth always hated the ocean and his forces eschewed any kind of sea warfare. Therefore, the navies of Círdan saw little action. And, for another, the armies of the Noldor were much stronger and much more actively involved in the events of the war.
Círdan later fled to the mouths of the Sirion river, and it was there where he was probably waiting when the War of Wrath was fought, Morgoth was overthrown, and Beleriand ended up being destroyed and drowned. He, however, along with many of his folk, escaped.
In the second age of middle earth, Círdan and his people settled in Lindon, an area that was what remained of Beleriand. They built the Grey Havens there, a great shipyard, where Elven ships would set sail for Númenór. His ally, Gil-Galad, the High King of the Noldor, set up a kingdom nearby. During the events of the second age, Círdan still managed to muster enough military strength amongst his people to resist Sauron, when Sauron invaded Eriador. Because of his high stature amongst the Elves in Middle Earth, Círdan was also given one of the Three Elven Rings, specifically Narya, the Ring of Fire.
During the events that ended the Second Age, Círdan was a key member of the Last Alliance of Elves and Men who struggled against, and overthrew, Sauron. He personally witnessed the combat betweenn Gil-Galad, Elendil and Sauron that left all three dead. He and Elrond together urged Isildur, the son of Elendil, to destroy the One Ring, but Isildur would not listen to him, and kept the Ring for himself.
During the third age, Círdan continued to live at the Grey Havens, building ships for Elves who had grown weary of Middle Earth and wished to leave. It was said that he himself would leave on the last ship to leave Middle Earth. His other significant deed during the Third Age was to hand over Narya, the ring of Fire, to the wizard Gandalf, telling him he would need it to keep his spirit burning during his journeys.
There were no longer enough Teleri left in Middle Earth for Círdan's people to have any military power. Of course, for most of the Third Age, this wasn't necessary, since Sauron was either in hiding or far away in Mordor for most of this time. Círdan did, however, take at least a counselling part in the last events of the third age. He, did, for example, send someone to the Council of Elrond to give advice. He may have been a member of the White Council. However, in the great events of the War of the Ring he is strangely silent.
At the very end of the Lord of the Rings, when the Ring Bearers and the major Elves are departing across the ocean, it is Círdan that meets them and sends them off in the ship he has prepared for them. He himself, however, doesn't leave, preferring to stay in Middle Earth until the last ship has set, and for all we know, he is waiting there still.
Círdan has very little dialogue even in Tolkien's unpublished works, and we don't know about what he was like as a character... just that he was wise and patient. Some could see this as a characteristic lack of characterization on Tolkien's part, while others could say it is Tolkien keeping some things in his work distant and understated, much as if the lives of the Elves were seen in the moonlight that they hold so dear.