New Netherland was not one for representative government. At least three colonial legislatures were established: the Twelve Men in 1641 (elected by an assembly in New Amsterdam to authorize a war against the local indigenous population, forcibly dissolved by Director Kieft when they started complaining about the authoritarian nature of the government), the Eight Men (elected by "the people" in 1643, presumably the same franchise as in 1641, and would fizzle out after a year or so), and the Nine Men (chosen from 18 candidates chosen by the people, fizzled out after Director Stuyvesant neglected to appoint any replacements), none of which were able to assert any sort of democratic foundation for the colony despite the Netherlands being one of the foremost republics on earth.
Ultimately, it would be until well after the British conquest and renaming to New York for a democratic legislature to take hold. Initially, the governor and council wrote laws, which would be approved by the Duke of York who owned the colony, but after governor Sir Edmund Andros was sacked for a number of offenses (including having the governor of East Jersey arrested, indirectly causing his death), Thomas Dongan was appointed and, in 1683, instructed to call for the election of an assembly of 18 members, apportioned as follows.

Counties did not exist, with the greater part of the colony divided into three ridings whose Long Island sections would become the three initial counties there. The borders of New York were not yet very well demarcated, and have been left off. Of interest are the seat given to Martha's Vineyard and Nantucket and the seat given to the holdings around Pemaquid in what is now Maine; these would be given to Massachusetts upon its rechartering in 1691.
Elections were conducted directly by freeholders in the more compact areas where everybody could be reasonably expected to turn up in one place, while in more scattered districts such as Martha's Vineyard and Nantucket there was a layer of indirect election. Tragically, the names of those elected have been lost, as have the proceedings of the session.
The assembly of 1683 passed a charter of rights which served as a constitution for the Province of New York. Dongan passed it, and it was sent to London where the Duke of York signed it in late 1684, after which it was sent to the Crown to be formally approved by Charles II. Ultimately, Charles II never got around to it before dying; the Duke of York assumed the throne as James II and proceeded to veto the charter that he had sent to himself for being too liberal. He instructed Dongan to dissolve the assembly of the now royal colony, though this did not reach New York until after the first session of the second assembly, the first assembly being dissolved upon the death of Charles II just to make sure everything was still above-board.
In 1688, James II would shove New York into the Dominion of England, chaired by the colony's former governor Edmund Andros. New English government over New York would not last long, however, as when news came that James II had been deposed in the Glorious Revolution, the Dominion of New England immediately collapsed into rebellion. New York was headed by a government helmed by self-proclaimed acting Lieutenant Governor Jacob Leisler, who, while awaiting a proper governor sent by William and Mary, summoned the first intercolonial congress in order to coordinate military actions against the French, who happened to be fighting a war with Britain at the time.
In 1691, Henry Sloughter, the newly-appointed governor, showed up in New York, where Leisler cordially handed over command of the colony and was promptly arrested and hanged for treason for his troubles. Only then could the colonial government resume itself in the manner in which it was run until the Revolution.