A grapefruit is not related to grapes, does not taste much like grapes, and is far bigger than any grape. The name is more than two centuries old, and the reason for it is still argued over.
The fruit itself is a Caribbean invention, and an accidental one. It arose as a chance hybrid of cultivated citrus, a cross between the sweet orange and the pomelo. Sources disagree about where. World Wide Words places its appearance in Barbados in the mid-1700s. Etymonline points to Jamaica, noting that a fruit of this kind was recorded in Hans Sloane's 1693 catalogue of Jamaican plants. In 1750 the Reverend Griffith Hughes described it in Barbados, where it was often called the "forbidden fruit." People of the time wondered whether it was the fruit of the Garden of Eden.
The word "grapefruit" first appears in 1814, in John Lunan's Hortus Jamaicensis, a botanical survey of Jamaica. Lunan gave a reason: the variety was called grape-fruit "on account of its resemblance in flavour to the grape." World Wide Words rejects that explanation.
The explanation usually preferred is the way the fruit grows. Grapefruit hang in dense clusters, and when small and green the clusters can look a little like bunches of grapes. World Wide Words admits this takes a vivid imagination. Etymonline lists both ideas, taste and clusters, and adds a third possibility: that "grapefruit" was partly a marketing name. The fruit had other names before it, including shaddock and pomelo, and grapefruit is the one that stuck.
For a long time the tree was grown mostly as an ornamental. Etymonline says the fruit was not widely eaten until the late 19th century.
The word grape has its own odd story. It comes from Old French grape, a bunch of grapes, probably from a verb meaning to pick grapes with a hook. English replaced its native word for the fruit, winberige, "wine berry," with the French one.