The Dutch colony of New Netherland became the English colonies of New York and New Jersey through military conquest and through treaties.
New Netherland was created as a Dutch colony in the 1610s. At this point, there was very little British presence in what is now the United States. Over the next 50 years, the British presence grew. When the Restoration of the Stuart monarchy happened in 1660, the British became more aggressive about colonial expansion and consolidation in North America. This led to a British expedition in 1664 that attacked and conquered New Netherland. The newly won territory became the colony of New York, which later split to become New York and New Jersey.
From 1664, these territories remained British almost continually. They were briefly retaken by the Dutch but the colony was then returned to Britain by treaty. From 1674 on, what had once been New Netherland was to be British on a permanent basis.