There are three levels of Federal disaster declarations that impact which disaster aid programs are made available to individuals in a disaster impacted area. (https://www.fema.gov/disaster-process-disaster-aid-programs)
The initial First Response to a disaster is the job of local government's emergency services with help from nearby municipalities, the state and volunteer agencies. In a catastrophic disaster if the governor requests, federal resources can be mobilized through the U.S.Department of Homeland Security's Federal Emergency Management Agency (FEMA) for search and rescue, electrical power, food, water, shelter and other basic human needs.
It is the long-term Recovery phase of disaster which places the most severe financial strain on local or state government. Damage to public facilities and infrastructure, often not insured, can overwhelm even a large city.
A governor's request for a major disaster declaration could mean an infusion of federal funds, but the governor must also commit significant state funds and resources for recovery efforts.
A Major Disaster can be a result of hurricanes, earthquakes, flood, tornados or major fires; the President then determines warrants supplemental federal aid. The event must be clearly more than state or local governments can handle alone. If declared, funding comes from the President's Disaster Relief Fund, managed by FEMA and disaster aid programs of other participating federal agencies.
A Presidential Major Disaster Declaration puts into motion long-term federal recovery programs, some of which are matched by state programs and designed to help disaster survivors, businesses and public entities.
An Emergency Declaration is more limited in scope and without the long-term federal recovery programs of a Major Disaster Declaration. Generally, federal assistance and funding are provided to meet a specific emergency need or to help prevent a major disaster from occurring.