Many expatriates maintain an international health insurance policy throughout their time abroad. This entitles them to scheduled and emergency health care abroad, usually in the closest European capital with a hospital approved by the insurance company.
However, if you cannot afford such a policy you will be reliant on the local healthcare service. The general opinion is that there are many good doctors in Bulgaria but the healthcare system as a whole, and some of its practises in particular, are outmoded and not very 'patient friendly'. Certainly those doctors that have studied or worked in the West will be a better bet, and many speak foreign languages. Such doctors can be found working in private health centres, and will often also work in a state hospital. They can also refer you to a state or private hospital.

If you are involved in an accident then you will be taken to Pirogov Emergency hospital (in Sofia). Here the doctors are competent  even if conditions aren't great and you may have to get a relative or friend to bring in supplies for you.
Not all countries have reciprocal health care agreements with Bulgaria, and you should check whether you are covered. If you live in Bulgaria, you or your employer should be paying Bulgarian health insurance for you and your family, which entitles you to emergency treatment.