most often, nothing.
it depends on the mistake, how egregious it was, and how well/poorly the physician communicated it with family.
drop a lung with a CVL? explain to patient, put in the pneumodart/chest tube, admit. be M&M'd, usually that's the end of it.
cannulate the carotid? explain to patient, hold pressure/call vascular, document, end of it.
miss something important and discharge the patient? call them back, explain, communicate.
order wrong drug to wrong patient? explain what happened, communicate.
The best thing you can do in these scenarios is directly communicate with patient and be honest.
The majority of medical errors will not significantly impact your career as a physician. However a pattern of negligent behavior and patient safety concerns will often result in probation from a hospital or revocation of privileges.
You have to be pretty outside the norm to have your license revoked for a medical mistake. Common reasons for license revocation: sleeping with patients, repeated drug abuse, medicare fraud, gross malpractice (ie, a family physician doing breast augmentation with only a nurse sedating resulting in a patient death, etc). Honest medical errors are expected and understood provided you communicate well and are otherwise a competent provider.