Most professionals today carry liability insurance to reduce their exposure to potential losses due to errors, omissions or bogus lawsuits. Surgeons and doctors are no different. The kind of insurance they carry is a specialized form of professional liability insurance called Medical Malpractice Insurance. It is usually very, very expensive because the potential of loss is so high.
If a client feels that the provider has made serious mistakes that have affected them personally, they sometimes sue for damages. From the provider's perspective, they have the insurance in place to prevent personal losses against that risk.
It's professional liability - it provides defense costs for the doctor, if they make a bad judgement call.
It IS expensive, most doctors pay somewhere in the six figure range, PER YEAR.
A patient cannot "file a claim" on it, they actually have to hire a lawyer and SUE the doctor. The insurance company won't take a claims notice from anyone EXCEPT the insured.