Please help! What is the name of the local anesthetic in needle the dentist uses before working on teeth?

Many names. However they all do the same thing. Some are lidocaine, septocaine, mepivacaine, carbocaine. They all come with different amounts of Epinephrine (0-1/20k) to make the anesthesia last longer.
Several. There are several types of local anesthetics used in dentistry. The most common these days are Lidocaine or septocaine, but there is also mepivicaine (aka carbocaine), which don't have vasoconstrictors and marcaine, a long acting anesthetic, just to name a few. If you ever have a reaction to an anesthetic, it is important to know which was used on you, so be sure to ask. Hope this helps.