What exactly is an empire, and when does a city-state, kingdom, or republic become one?

An empire is a kind of political organization in which a small group of people dominate a larger group of people. City-states, kingdoms, or republics can become empires when they acquire territory that's not incorporated into their original borders. You'll have what political scientists call a "center" and a "periphery." The center is the original city-state, kingdom, or republic, and the periphery is the collection of conquered lands. Think of the Roman Empire or the British Empire: the centers in those cases would be the city of Rome and its hinterland and the United Kingdom, respectively. The peripheries are what we usually think of when we hear the names of the empires: Gaul, Hispania, or Anglia in the Roman example; India or Australia in the British.
Those centers and peripheries are the hallmarks of empires, but they're not the only things you need to maintain them. A critical component of any empire is the cultural will to keep it going. The people of the center, the imperialists, have to believe in their empire, and they have to maintain a culture that makes it acceptable to dominate whole peoples and subjugate new lands. Everything from art to literature to public policy to sport is enlisted in the creation and maintenance of an imperial idea. To better understand this, you should read Edward Said's excellent book on the subject, Culture and Imperialism.
There's a lot more to understanding the origins of empires. One of the best things you can do if you want to learn more is to read books written by people who created empires. History books about empires are fine, but you'll really understand the impulse to dominate whole populations if you get inside the heads of conquerors. The Gallic Wars by Julius Caesar is an excellent place to begin. The Lake Regions of Equatorial Africa by Richard Burton is a classic of imperialist exploration.