Native Americans, including the Lenape Indians, inhabited the area long before Europeans arrived.
In 1524, the first European to the area was Giovanni da Verrazzano, an Italian sailing for the French. The French never colonized the area.
English explorer Henry Hudson rediscovered the mouth of the Hudson River in 1609 and explored Hudson Bay in 1610. Like other explorers, Hudson was actually looking for a shortcut to Asia. He did not found any settlement there.
French explorer Champlain mapped the east coast of America from Cape Bretton, Nova Scotia, to Cape Cod, Mass in 1607, not quite far enough south to find (or found) New York, but close.
Peter Minuit was a German but was a director of the Dutch West India Co. He reached Manhattan Island in 1624 and purchased it from the Indians with beads, ribbons, and trinkets valued at the amount of 60 guilders, or about $24.
This became the colony of New Amsterdam (Niewe Amsterdam) in 1625. That is the date that appears on the Seal of the City of New York.
New Amsterdam separated the English colonies of New England from the other English colonies in the south. Clashes between the Dutch and the English were inevitable. England's Charles II claimed all the Dutch land, and in 1664 he gave it to his brother, the Duke of York.
The Dutch weren't prepared to fight the English, so in 1664, New Amsterdam became New York.
New York City as we know it (Manhattan, The Bronx, Brooklyn, Queens and Staten Island) was founded in 1898. Before they were just separate boroughs, each with their own jurisdiction.
The state of New York was founded in July 26, 1788.