I'll describe how healthcare is organized over here (NL). It is definitely not the same all over Europe!

1. Health Insurance
Everyone is obligated (by law) to have health insurance. If you're not insured, you will be fined and you have to pay for the months that you went without insurance.
Health insurance is between EUR 90 and EUR 140 per month. It depends on how much coverage you want (especially extended dental care can increase the premium).
If your income is below a certain figure, you get an extra allowance from the government to compensate.

2. Dental care
Most people have a dentist, who checks their teeth every six months. This is covered by insurance.
If you need dental work done, payment depends on what has to be done and what your coverage is. Part of the bill will always be for yourself. Preventive dental care is usually covered by insurance.

3. GP (family doctor)
Most people have the same GP for years. All consultations are free of charge.
Appointment to made by phone; appointment usually within 3 working days. Sometimes, when it's busy, they offer you a quicker appointment with another doctor.

4. Physiotherapy/Ergotherapy/....therapy
You can make an appointment directly with therapist, without any paperwork from a doctor. The first 9 treatments/sessions are free of charge. Depending on your coverage, you have to pay for it after that or not.
(I'm not sure how it works if you need physiotherapy for two different ailments in 1 year.)

Questions?
More to follow later.