There are two aspects to this question: the meaning of the word "empire" and the various traditions and rules around the titles of monarchs (king versus emperor in this case).

The English word "empire" has two meanings: it can mean either "large, colonial state", or "state ruled by an emperor". It originates from the Latin "imperium", which originally just meant "command", but also came to refer to the Roman Empire ("Imperium Romanum" in Latin). So, there we have the first and prototypical example of an empire: the Roman Empire. It is important to note that the Roman Empire was called so already when it was ruled as a republic. So the name "Empire" does not, originally, refer to a specific form of government. Later however, the Roman Empire came to be ruled by emperors ("imperatores" in Latin), and so the term empire came to be associated with emperors, even though there was originally no connection.

Since the Roman Empire was a large, powerful, colonial state, we get the first English meaning of the word "empire". The British Empire was just such a large, powerful, colonial state. So there was never a necessary connection between empire and emperors; one should probably also point out that the British Empire never had this name officially. This was more of a way to stress that one was talking of the whole of the empire including the colonies instead of just the British Isles. Officially, it was just called a "kingdom", for example during the 19th century the official name was the "United Kingdom of Great Britain and Ireland".

The other aspect of the question is: what is the difference between a king and an emperor? By the peak of the British Empire, in the 19th century, this distinction was basically just a matter of tradition. Some monarchs were traditionally called kings, others were emperors, that was it; there was no real difference except that emperors ranked slightly higher in matters of protocol. The origin of the tradition goes back to the Middle Ages. At the time, there was a belief that there were many kings, but only one emperor -- the emperor being the secular head of the Christian world, corresponding to the Pope, who was the spiritual head. The title of emperor was usually held by whichever king controlled Italy or Rome, since the medieval emperors were seen as successors of the Roman emperors. That said, even in the Middle Ages, it was mostly an empty honor -- few other kings actually considered the emperor their superior, as theory would have demanded.

Then you might ask, why were there multiple emperors in the 19th century? Simply because multiple kings, through various reasons, claimed to be the successors of the Roman emperors. The emperor of Austria was the dynastic successor of the Holy Roman Emperors, who controlled Rome in the Middle Ages. The German Emperor also claimed to be successor of the Holy Roman Emperors, because their territory had been mostly Germany. The Russian tsar claimed to be successor of the Eastern Roman emperors, seated in Constantinople.