Had a molar root canal this morning a small 1mm piece of file broke off in the one root when they tried to keep drilling to finish it perforated the tooth. He finished the other 2 canals and put a temp filling in and showed me the x-ray and what happened. He said to go to my reg dentist to get an opinion weather to extract it or possible surgery (very expensive)to fix it. He said I didn't have to pay for the visit. What options do I have ? What is mta and how expensive is it ?

The best kind of dentist to see now is an Endodontist(specializes in root canals). This specialist will be able to give you the best prognosis for the tooth. Perforations can often be sealed allowing you to keep the tooth but you really need to see someone who is experienced with them.

MTA is probably Mineral Trioxide Aggregate

See Mineral Trioxide Aggregate Review