The statues were carved by the Polynesian colonizers of the island, mostly between circa 1250 A.D. and 1500 A.D.
At some point after the 1722 Roggeveen visit, all of the moai that had been erected on ahus were toppled, with the last standing statues reported in 1838 by Abel Aubert Dupetit Thouars, and no upright statues by 1868, apart from the partially buried ones on the outer slopes of Rano Raraku.
Website with more info on the history of the Moai
Moai History