Number theory (or arithmetic) studies numbers, especially integers. Typical questions are the distrubution of prime numbers and the study of integer (or rational) solutions of algebraic equations with integer coefficients, also called Diophantine equations.
(Notice that by a theorem of Matiyasevich (spelled also Matiyasevich), for every statement in mathematics (say ZF set theory) there is a Diophantine equation whose solvability is equivalent to the validity of the statement. Of course, that does not mean that addressing a problem as a Diophantine equation helps in solving it. )
The analytic study of the asymptotic behaviour of the distrubution of prime numbers on the positive integer line is the main subject of analytic number theory; it also studies the distribution of rational numbers with small denominators.
A roots of an algebraic equation with integral (or equivalently, with rational) coefficients is called an algebraic number. Extending the field $\mathbb{Q}$ of rational numbers by abstract solutions of such an algebraic equation in a minimal way leads to an algebraic extensions of rationals, so called number field. There is a number of algebraic structures related to the study of algebraic integers and number fields in particular; these questions comprise algebraic number theory including its central part, the class field theory. The Galois theory is one of the principal ways of study of such questions. Algebraic geometry is very effective in the expression and study to more elaborate questions in this study which in its historical development brought up very many major concepts of algebra and algebraic geometry including central notions like a group, ideal, and Grothendieck motive.
It is one of the oldest branches of mathematics and very popular as many difficult problems can be stated in a rather elementary and simple manner, e.g. the Fermat’s Last Theorem and Riemann hypothesis.
Weil conjecture, Fermat's last problem?, abc conjecture
excact real computer arithmetic?
The following survey of Connes-Marcolli work has an accessible quick introduction to algebraic number theory