When a person dies without a valid will in New York, their estate is subject to the laws of intestacy. In such cases, the court appoints an administrator to handle the estate administration process. The administrator's responsibilities include identifying the deceased person's rightful heirs according to the state's laws, gathering and managing the assets, paying debts and taxes, and distributing the remaining assets among the heirs based on the statutory rules of distribution. The process aims to ensure a fair and equitable distribution of assets in the absence of a will.