When you see EMS on ambulances the EMS stands for "Emergency Medical Services" which is what the EMT people are providing.
The EMT are the people who are Emergency Medical Technicians.
EMT or emergency medical technicians are medical technicians that are certified to provide basic emergency services (as cardiopulmonary resuscitation) before and during transportation to a hospital.
The EMT works for the EMS and an EMT is trained to provide medical care but is less trained than a Paramedic is.
EMT people are usually the first responding to arrive to asses the emergency and provide initial treatment while the Paramedics who are trained more and can provide more care are on the way when needed to take over and transport you to the hospital.
Paramedics have more skills and go through more training to become a paramedic and can provide more care to the patient.