This name was borne by the heroine of Lewis Carroll's novels Alice's Adventures in Wonderland (1865) and Through the Looking Glass (1871).
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.