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.
Among Christians, this name was originally more common in Eastern Europe. It was borne in the 12th century by Saint Elizabeth of Hungary, a daughter of King Andrew II who used her wealth to help the poor. In medieval England it was occasionally used in honour of the saint, though the form Isabel (from Occitan and Spanish) was more common. It has been very popular in England since the reign of Queen Elizabeth I in the 16th century. In American name statistics (as recorded since 1880) it has never ranked lower than 30, making it the most consistently popular name for girls in the United States.
Besides Elizabeth I, this name has been borne (in various spellings) by many other European royals, including a ruling empress of Russia in the 18th century. Famous modern bearers include the British queen Elizabeth II (1926-2022) and actress Elizabeth Taylor (1932-2011).