The most common type of surgical error is giving too my anesthesia.
If a person is given too much anesthesia then it can be deadly and the person may not wake up from the anesthesia.
However it's rare for someone to die of too much anesthesia but it has happened.
Other surgical errors that are common are leaving surgical tools behind or sponges.
Those are less serious but can cause pain and even serious problems until the tools or sponges are removed.
Some people will sue a hospital for malpractice if they do have a surgical error.
A malpractice lawsuit is a Civil Lawsuit.
When you sue someone for malpractice you go to a Civil court proceeding and sue them for the malpractice.
For a malpractice lawsuit the alleged victim of the malpractice lawsuit brings the lawsuit for the purpose of remedying perceived damages sustained via monetary compensation.
There is no state involvement in the lawsuit unlike a criminal lawsuit.
Malpractice itself is not a crime so it would not be a criminal lawsuit or criminal court proceeding.
You would need to find a lawyer that deals in Civil or malpractice lawsuits.
It can be hard to sue for malpractice but it can be done if you have enough evidence.