All the definite 'sightings' of Thorondor take place during the First Age. We first encounter him helping Fingon rescue Maedhros from Thangorodrim when the Age had hardly begun, and he survived at least long enough to take part in the War of Wrath at the end of the Age.
Thorondor's ultimate origins are uncertain, but he seems to have existed before the beginning of the First Age - perhaps a very long time before. The Silmarillion tells us that the race of Eagles were sent by Manwë to watch over the Noldor who had travelled back to Middle-earth. Earlier versions of the story published in The History of Middle-earth state more explicitly that Thorondor was one of these, and that he even remained in contact with Manwë during the First Age.
Thorondor's fate is equally uncertain. The only clue is a passage that appears in The Lord of the Rings, referring to '...old Thorondor, who built his eyries in the inaccessible peaks of the Encircling Mountains when Middle-earth was young.' It's plain from this that Thorondor departed Middle-earth some time after the end of the First Age. Perhaps the greatest Eagle of them all simply died of old age, but his mighty body carried a spirit sent by Eru, so perhaps, just possibly, he flew back to his master's halls on Taniquetil and still soars in the airs of the Undying Lands.