While this particular article does not address the specifics for a New York divorce (there is no Summary Divorce and no Arbitration for divorce in NY), it does outline the nine different types of divorce, depending upon how complicated the situation is.
Contested & Uncontested: To obtain a divorce you must commence a lawsuit or "action" against your spouse for divorce. If that action moves forward where the spouse or "parties" fight over every issue that must be decided in a divorce (distribution of marital assets, child custody and visitation, etc.), then this is a contested divorce. An uncontested divorce is where the parties settle their lawsuit, or even better, use alternative methods wherein they reach an agreement on all issues before they even file for their divorce, and then these parties DO NOT step into the courthouse...instead their paperwork goes to court.
What are the alternative methods?
Mediation is a process in which an impartial third party meets with a couple to help them reach a mutual and informed agreement for the terms of their separation and/or divorce. https://www.spiriolaw.com/Practice-Areas/Divorce-Mediation-Long-Island
In the Collaborative Process the attorneys and parties agree that they will not go to Court and will instead focus their energies on settlement. The parties can therefore control certain aspects of their divorce or dissolution, such as the cost and the timing. https://www.spiriolaw.com/Practice-Areas/Collaborative-Law-Attorney-Long-Island
New York is now a No-Fault Divorce state, which means you do not have to allege or prove fault grounds in your action to obtain a divorce.
Finally, same sex marriage is now legal nationwide and divorce is governed by state law.
Selected excerpt(s), photo and linked article courtesy of PRFIRE.