"Number theory," he said, "is the theory of numbers."
In number theory, the set of numbers in question is often the rationals or naturals. Real or complex numbers are the work of real or complex analysis. Primes are of particular interest, and a lot of work in number theory is the study of primes and how to prove that a number is or is not prime (see also: The Node of Prime).
The fundamental theorem of arithmetic says that every natural number is uniquely expressible as a product of powers of distinct primes. This, along with the concepts of divisibility (see also division algorithm) and factorization, are the initial foundations that are often laid in any number theory course.
Built upon these basics are the Euclidean algorithm, studies of linear Diophantine equations, congruences, systems of congruences (a cousin to systems of linear equations), the Chinese Remainder Theorem, multiplicative orders, and many more. Aside from divisibility and factorization, congruences are the most important basic ground to cover; every congruence is a simplified form of a linear Diophantine equation (notationally), and congruences can be shown to have all the properties of equations, aside from some trickiness when it comes to division.
The major branches of number theory are analytic and algebraic, e.g., the prime number theorem is an analytic result, whereas Fermat's little theorem is an algebraic result.
Some well-known results of number theory
Source: the number theory course I took, whose book was:
Holt, Jeffrey J. Discovering Number Theory. W. H. Freeman: New York. 2000.