The Roman Republic was gone. In its place, the government was now a dictatorship, under the leadership of an emperor. The first Roman emperor was Augustus, the adopted son of Julius Caesar. He was emperor for 45 years. This was the beginning of the Pax Romana or Roman Peace. This is a very misleading title. There wasn't really peace.
Rome continued to expand the empire, mostly through conquest. Rome itself was still beset by criminals and sometimes riots. It is called the Pax Romana because the empire itself was stable. The people knew that there was an emperor to run things and Roman legions to take care of wars and riots. They didn't have to worry about somebody coming in and conquering Rome and destroying their beautiful city.
Many of Rome's most spectacular and huge construction projects were built during this time period, the period we call the Roman Empire. Thousands of miles of roads were built to keep the empire united. Art, literature and theatre flourished and grew. Rome was at its height.