I am not clear about national american health system and private health care and insurance - how would i pay for a physiotherapy appointment?

Many possibilities: There is no national health system in america, unless you have medicare. If you have private insurance it will generally cover a portion of the cost of a participating doctor's visit and you pay the rest (called a co-pay). Co-pays are generally higher when you see a specialist. If the doctor or therapist does not participate, you pay the entire bill. Your insurance may reimburse part of the cost.

Depends: It depends on your insurance. Each insurance company has it's own rules about what is considered "covered" services. Even if they cover a treatment, you still need to find out what percentage of the bill they pay. Also, they often require a doctor to prescribe physiotherapy before it will be covered. It is important to know the guidelines of your personal insurance plan so you know what to expect.