Why was the colony of New York created?

The Dutch wanted to expand their land development and trade in order to compete with England, France, and Spain.

Why? To make money...

The Dutch (or Nederlanders, or Walloons, or whatever...) established "New Amsterdam" (as "New York" was originally known) as a trading post and commercial colony. It was no nation, but rather a private corporation, the Dutch West Indies Company, that wanted to set up colonies as, you might say, franchises: people who wanted to try a new life, could go to one of their pre-established colonies, and give it a shot. They would then support industries in which the DWIC was engaged.

The business in New Amsterdam was beaver pelts. This was a very popular material in Europe, particularly for hats (beaver is somewhat waterproof) and the Americas had a huge supply. Trappers would come down the Hudson and other rivers with beaver pelts, and trade them for goods. The DWIC would take their cut, and all would be happy.

Problem was, the beaver trade was not lucrative enough to cover the cost of running and protecting a distant colony. So, when the British threatened to take New Amsterdam by force, the Dutch simply gave it up. As to why the Bitish wanted it, it had a good port, and the Dutch were in the way of their consolidating the region under their control.

The beaver trade, while gone, is not forgotten. You can still find Beaver Street down in lowest Manhattan, and a beaver is featured on the flag of New York.