Number theory, one of the branches of pure mathematics, is the study of numbers and their properties. In many other branches of mathematics, numbers are used strictly to convey information (e.g. 75%, 3 apples, 2x² - 1). Number theory, on the other hand, studies numbers (usually the integers or some subset thereof, like the natural numbers) for their own sake. The properties of numbers are indeed fascinating.
That's not to say that number theory has no practical value. Number theory is of great value in the information age. Knowledge of congruences and prime numbers has been invaluable in the areas of error-checking and -correcting codes and cryptography, respectively.
The first number theorist was probably Pythagoras, who believed that "all is number". The later Greeks, who were no doubt influenced by him, valued number theory (which they called arithmetic) above arithmetic (which they called logistic). The Greeks' mathematical discoveries were a cause (or perhaps a symptom) of their contempt for practical mathematics.
Number theory is a fairly wide branch of mathematics. Here are some pages on Math Lair that discuss topics related to number theory: