Four of the 50 states in the U.S. call themselves a Commonwealth - Virginia, Massachusetts, Pennsylvania and Kentucky. Constitutionally, there is no difference between a state and a commonwealth.
Commonwealth originally meant a region governed by the people, not a monarch. England was a commonwealth from around 1649 to 1660. During the American Revolution, the colonies of Massachusetts, Virginia and Pennsylvania declared themselves commonwealths. Thereby they signaled that they were no longer governed by the British monarchy but were an independent republic. At the time, Kentucky was part of Virginia. In 1790 when it separated from Virginia, Kentucky chose to retain the commonwealth moniker.
During the Civil War, Virginia seceded from the union to become a confederate state. West Virginia seceded from Virginia and decided to join the union. At the time, West Virginia did not choose to retain the commonwealth status; instead, it became a state in the U.S.A.
U.S. territories of Puerto Rico and the Northern Mariana Islands also use the "commonwealth" moniker. These regions do not have some of the same rights as states - most notably that of having representation in the U.S. senate.