They hoped that Jesus would say either "yes" or "no." If he said "yes," then the Jewish people would be angry with him for telling them to pay taxes to a foreign government. If he said "no," then the religious leaders could tell the Romans that Jesus was teaching the people to break the Roman laws.
Because Caesar was the ruler of the Roman government, they could refer to the Roman government by Caesar's name. (See: [[rc://en/ta/man/jit/figs-metonymy]])