An integer is an element of $\mathbb{Z}$, which may defined as the free group on one generator or as the initial ring.
As a group, $\mathbb{Z}$ is abelian and is the Grothendieck group of the monoid (or semigroup) $\mathbb{N}$ of natural numbers.
The group of natural numbers is naturally even a rig – in fact the initial rig – and this multiplicative structure extends to $\mathbb{Z}$ to make it a ring – in fact the initial ring.
The underlying sets $\mathbb{Z}$ and $\mathbb{N}$ are isomorphic. Some subcultures of mathematics (and not only set theorists) use the term ‘integer’ synonymously for a natural number. Computer scientists distinguish between ‘unsigned integers’ (natural numbers) and ‘signed integers’ (integers as described here). Translations can also cause confusion with the term ‘whole number’.
In number theory, one generalises integers to algebraic integers, an instance of the red herring principle. Accordingly, some number theorists will call the integers ‘rational integers’ to clarify; $\mathbb{Z}$ is the ring of algebraic integers in the number field $\mathbb{Q}$ of rational numbers. (Compare, for example, Gaussian integers and Gaussian numbers.)
The symbol ‘$\mathbb{Z}$’ derives from the German word ‘Zahlen’, which is a generic word for ‘numbers’. (Compare Dedekind's use of that word in the title of his famous book on the foundations of real numbers.)
geometry modeled on the formal dual of the ring of integers is arithmetic geometry
A formalization in terms of homotopy type theory, using a unary notation, is in
(A different common formalization of integers in type theory is in a binary notation, as in the Coq standard library. Binary notation is exponentially more efficient for performing computations, but the unary notation was convenient for calculating $\pi_1(S^1)$.)