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 name (in various spellings) has been common among European royalty, being borne by ruling queens of Naples, Navarre and Castile. Another famous bearer was Joan of Arc, a patron saint of France (where she is known as Jeanne d'Arc). She was a 15th-century peasant girl who, after claiming she heard messages from God, was given leadership of the French army. She defeated the English in the battle of Orléans but was eventually captured and burned at the stake.
Other notable bearers include the actress Joan Crawford (1904-1977) and the comedian Joan Rivers (1933-2014), both Americans.
The name was very popular until the 15th century in England, usually in the vernacular form Maud. Both forms were revived by the 19th century. This name appears in the popular Australian folk song Waltzing Matilda, written in 1895.