There are mutiple phases of disaster response and relief that involve different services and skills. At the moment of a disaster, and depending upon what type of disaster has occurred, emergency response agencies may have little or no time to prepare before the crisis starts, or they can have days to weeks to prepare, for example, in the case of hurricanes being tracked as they slowly approach land. As the crisis hits, the first emergency service personnel to respond and arrive on the scene are what are called "first responders," police and fire departments, ambulence crews, and possibly hazardous response units. These departments or crews require a high level of training in on-site medical care, crowd control and skill in cordoning off affected areas to contain damage and deter looting and, in the case of a biological weapons attack, prevent potentially affected people from fleeing the scene and infecting additional citizens.
Additional emergency services involved during the course of a disaster involve the same agencies -- police, fire departments, emergency medical personnel -- but expand to include hospitals, use of large cavernous buildings like gyms manned by Red Cross personnel to provide shelter to the newly homeless, and additional security personnel, for example, mobilization of the National Guard to maintain order and help with the wounded and extraction of individuals -- dead and alive -- trapped under fallen debris.
Once the initial crisis has abated, expansion of services to care for the wounded and homeless continue to increase. Additional rescue units are brought in to locate and extract wounded from trapped situations, and emergency service personnel from outside areas are brought in to help clear rubble, locate any still missing bodies, and begin rebuilding essential facilities and shelters. The Red Cross effort at helping affected families increases in direct proportion to the scale of the disaster, and additional National Guard units might be mobilized if needed to help maintain order.
In the meantime, federal resources should be flowing in by now, including money to help affected families begin to rebuild their lives, Federal Emergency Management Agency officials to provide the President an assessment of the extent of the damage with an eye toward requesting additional resources. Depending upon the scale of the disaster, federal troops might be brought in to help construct infrastructure essential to the establishment of a functioning community -- for example, resetting pipelines, paving damaged roads, setting up emergency electrical generators, and so on.
These are the types of emergency services commonly deployed in the initial and later phases of a disaster.