What is legally considered hospital malpractice and how do you go about filing a lawsuit for this?

Courts have found hospital malpractice in a number of situations.  First, a hospital may be held vicariously liable for the ordinary negligence or malpractice of its employees, such as physicians.  N.X. v. Cabrini Medical Center, 97 NY2d 247 (2002) 

In order for liability to lie, the doctor must be an employee of the hospital, be carrying out the hospital's business, and acting within the scope of his or her employment.  A hospital may be held liable for the negligence of an E.R. doctor, where a patient checks into the emergency room for treatment, and the hospital holds itself out as a purveyor of health care services.  Hospital malpractice may also be found where there was a breach of the duty to use reasonable care in providing competent medical personnel.