In the US, there are a few different systems, depending on what state you live in.
In general, in an emergency, you dial 911 and state your emergency. The dispatcher will contact the appropriate people--you don't have to state which emergency service you want. You can ask for an ambulance, but the dispatcher will know who to call for that. Depending on where you live, ambulances may be free, or you may have to pay for your ambulance ride. Some insurance companies cover ambulances in an emergency; some do not.
EMS (Emergency Medical Services) vary widely from state to state--there is no Federal EMS. Some EMS departments are separate entities, some are made up of police and firefighters, some are completely volunteer, as are some fire departments.
Even within the same state, you will have cities with a police force, paid firefighters and a separate EMS with its own ambulances, and then smaller towns with no police force of their own who rely on county or state police/sheriff services, volunteer firefighters and volunteer EMS personnel who rely either on a paid ambulance service or, with an agreement with a larger city, the ambulances from that city.