A "hearing" is a time for the court to interact with the parties in a case before it on a specific date and time for a specific purpose. Some hearings are procedural and some hearings involve the taking of evidence. Unless specifically indicated to the contrary, hearings must be attended in person.
A "case" consists of all of the filings and proceedings assigned the same case number by the court and generally flowing from the same petition or other initial request for relief.
For example, a suit to collect child support is a "case", in which there might be several hearings and many legal documents filed, all flowing from a single initiating document.
The best procedure in a civil case would be to call the other side's lawyer, determine if they object to a continuance of the hearing, and if not, to contact the clerk of the court making the request and follow up with a written filing with a case caption with the court, with a copy mailed to all other parties in the case, requesting the extension of time. The Court may grant or deny it in its discretion, but if you can truthfully state that the other side does not object to the extension, it is very likely to be granted.
Family court in New York State has both civil and criminal jurisdiction. In criminal cases (e.g. domestic violence) the process may be somewhat different. This is not an area of law in which I practice, so I am not familiar with the process for continuing hearings in criminal cases.