In general, an "empire" is thought of as a political organization that comprises several nations or nationalities of people, while a "kingdom" or nation is more nationally/ethnically homogenous.

You can probably see that he definitions of those are very tricky, especially when we apply the notion of "empire" to the ancient world, which had very different understandings of race, ethnicity, and nationhood/nationalism.

The word "emperor" has roots in the Latin "imperium," which was the notion that the state had sovereignty over individuals (and I'm going to stop there, because it's been a good 15 years since I took a Roman history class).

To move to a period I'm more familiar with, there are a couple of good examples from the run-up to World War I (mostly just because I've been re-reading Robert K. Massie's "Dreadnought"): The Austro-Hungarian Empire was thought of as an empire because its emperor, Franz Joseph I, was the emperor of Austria, the king of Hungary, and had other titles regarding the empire's other component states. Victoria of Hanover was the queen of the United Kingdom and Empress of India (from 1876); her investiture with the title happened after the Mughal Empire was finally defeated. The British thought of the UK as a nation with a king (or queen), but their overseas territories as comprising their empire.

In practice, then, you can see that the title might be given to the same person, and that some person could have more than one title. So, it really depends on who's claiming what from whom, and the legal structure of the component nation-states.