As an English name it has been in use since the Middle Ages, originally in the form Clare, though the Latinate spelling Clara overtook it in the 19th century and became very popular. It declined through most of the 20th century (being eclipsed by the French form Claire in English-speaking countries), though it has since recovered somewhat.
This is also the name of an Irish county, which was itself probably derived from Irish clár meaning "plank, level surface".
As a rarer (but older) masculine name it is probably derived from a surname: see Fay 1 or Fay 2.
In the United States this name steadily grew in popularity through the 1950s and 60s. It shot up the charts after the release of the 1970 movie Ryan's Daughter. Within a few years it was in the top 20 names, where it would stay for over three decades. Famous bearers include the Canadian actors Ryan Reynolds (1976-) and Ryan Gosling (1980-).