New Amsterdam was a trading post founded by the Dutch in 1624. When the British took control in 1664, New Amsterdam was renamed New York after the Duke of York, who received the land as a gift from the King of England.
That's a bit difficult to say, since New Amsterdam was founded by the Dutch. This was the famous 'trade' of Manhatten for $24 worth of beads. My ancestors were early settlers in New Amsterdam, arriving there from the Netherlands in 1652.
New Amsterdam is the original Dutch name for the modern New York City. The town was founded in 1625. The Dutch name was Nieuw Amsterdam. It was part of the Dutch settlement of what became the state of New York, in the subsequent United States of America.