Prior to 1874, villages were incorporated as distinct municipal governments through a special action of the New York State Legislature. Following a revised State Constitution, Villages could no longer be incorporated in this way, but were to be incorporated through local initiative pursuant to the Village Law.

The below account of Village History has been provided by the Hastings Historical Society:
According to the local paper, The Statesman, people in Hastings-on-Hudson (link to Greenburgh 1867 Map) had been talking about incorporation since 1875. The place had already been called Hastings-on-Hudson (or a variation on that name) for at least 40 years. But incorporating as an official village would mean that the community could elect its own government instead of relying on Greenburgh for services. The paper was all for it. “Hastings looks for progress and increased prosperity by incorporation,” wrote a reporter, “which means better streets, increased value of real estate, and higher taxes.”
In 1876 the same paper reported that most people would, in fact, rather be annexed to the city of Yonkers.
But by 1879 the tide of opinion had turned. The prominent citizens of Hastings took up a collection to raise money to pay for a survey of the proposed area to be covered by the town, one of the steps necessary to apply for incorporation. In September the surveyor, George Wiley, was working on the map. A village census was also in progress, another crucial part of the application for incorporation. (The total number of inhabitants was 1,226.) In October, everyone was invited to come to Samuel G. Dorland’s general store to see the completed map (the original map is located at the Hastings Historical Society).
In the opinion of The Statesman, most citizens favored incorporation. However, there were some who thought the proposed village was too large. It was, after all, two square miles. “It takes in roads which are not needed, which will be expensive to work, and keep in repair,” the paper reported. A public vote on incorporation was held in Protection Hall on November 18th.
There were a lot of votes against incorporation (conflicting reports give the number as 221 to 209, or 104 to 75), but in any case the supporters won the day. And that is how we officially became Hastings-on-Hudson.

The Certificate of Incorporation is on file in the Westchester County Clerk's Office. The printed transcript can be found HERE.
The Notice of Incorporation in the first Board of Trustees Minutes Book includes the recorded votes for incorporation (104 to 75) and the votes for the first slate of Trustees.
The first two Village Board of Trustees meetings were focused on appointing the Village Clerk. John Leigh, Acting Village Clerk at the time, was appointed as Village Clerk.