New York has an extremely complicated and convoluted court system. Until such time as there is a constitutional convention (and I’m not holding my breath for that) we will have to continue to navigate this complex maze.
In order to make things more understandable for families and couples in dispute, the following is a summary of the courts that would address your problems:
Family Court is a court of limited jurisdiction, which means it only handles certain types of cases. Family Court handles the following matters:
- Juvenile Delinquency – This applies to a person between 7-16 who has committed an act that would constitute a crime if committed by an adult.
- Child and Spousal Support – This applies to both intrastate and interstate (Uniform Interstate Family Support Act) support cases;
- Paternity–Determining who is the father of a child;
- Termination of Parental Rights;
- Guardianship and Custody;
- Person in Need of Supervision – This applies to a person under 18 years of age who does not attend school, is incorrigible, ungovernable or habitually disobedient beyond the lawful control of a parent or who commits certain [minor] crimes;
- Abuse and Neglect; 
- Family Offenses; and
- Foster Care and related matters.
Supreme Court is a court of general jurisdiction. This means you would be in Supreme Court if you were in a car accident, had a breach of contract case, etc. In a family situation, Supreme Court has exclusive jurisdiction over divorce, annulment and separation proceedings. Only in Supreme Court can a marriage be dissolved and property distributed.
There is concurrent (or dual) jurisdiction in Supreme and Family Court for custody, and child and spousal support. Supreme Court can also issue Orders of Protection, as can the criminal and family courts.
HOW DO I KNOW WHICH COURT TO GO TO?
If you are not married to the other person, then Family Court is your only choice. If you are married and wish to dissolve the marriage or distribute property, you will be in Supreme Court. However, there are times that someone, for various reasons does not wish to terminate the marriage but is looking for support or to deal with a family offense situation or custody, which can then be dealt with in Family Court.
Confused? Join the club. There may be strategic reasons for putting a married person in a Family Court situation. While there has been a push to consolidate proceedings, it is possible to begin a support and custody proceeding in Family Court and then a divorce in Supreme Court and actually be in two courts at one time! Unfortunately, this is expensive, time consuming, and difficult for the Judges who may be lobbing overlapping issues back and forth to each other.
 Surrogate’s Court also has jurisdiction
 Potentially can be in a criminal court also
 See above
 For example, some counties have Integrated Domestic Violence Courts that would handle a divorce, family court, and criminal court proceeding at the same time.