What is the name of that liquid doctors use to clean the area on the patients body before they cut them?

Many Options. The most popular skin preps prior to a surgical procedure are povidone iodine (betadyne), Chlorhexidine (chloraprep), and isopropyl alcohol.
Betadine. I usually use betadine (an iodine based preoduct) but Chlorhexidine can be used as well. I have even done some preps with rubbing alcohol.