If I got a tooth removed and the dentist left a little piece of tooth, what are my legal rights?

Well first I hope that you have gone either back to the dentist or to a different dentist for help at this point in time.  How do you know it is a little piece of tooth?  Is this self diagnosing here that is going on?  Bringing an action for medical or dental malpractice in New York is not the easiest thing.  The law requires certification from another MD in the field that there is a valid claim for malpractice.  And there have to damages in order to make a claim.  But what is good is that consultations for personal injury actions are generally free of charge.  So I would first get to someone to help you with your mouth and then get to someone to consult with on the alleged malpractice claim.  Good luck.