The mere fact that you or someone in your family had what the doctors call a "bad result" does not mean that you can sue your doctor or hospital. But if the bad result is caused by negligence or by what the law calls "medical malpractice", you can sue and obtain money damages for your injuries. There are some things you should bear in mind when deciding whether you have a valid claim.
What is considered medical malpractice? Medical malpractice is defined as a deviation from accepted medical practice.
Under New York law you only have a short amount of time to bring a case for medical malpractice (this is called the statute of limitations). For most adults, this means that you have only two and a half years to start your lawsuit from the time the doctor made the error or failed to do what he was supposed to do (sometimes this is extended but under very limited circumstances). The time is even shorter (generally, a year and 90 days to start suit) if you are suing a municipal hospital or a doctor who works in one and even then you must first serve what is called a "Notice of Claim" within only 90 days!
So if you suspect that what was done to you or a family member was caused by a doctor's carelessness, the best thing to do is to see a lawyer who has experience in handling medical malpractice law suits as soon as possible.
An experienced lawyer will know from your story whether the matter is worth investigating. If the case seems like it might be a case of malpractice, the lawyer will retain a doctor to review your records and render an opinion as to whether you may have a valid claim.
Most reputable medical malpracticelawyers will not charge you anything for their time in reviewing your case. And most lawyers who practice in this field will charge a contingent fee once they decide to take the case, which means they will only get paid if they succeed in getting you compensation.