numbers $z$ such that $z^n=1$ for some natural number $n$; here usually $z$ is in $\mathbb C$ or some other field

An $n$-th root of unity is a complex number $z$ such that $z^n=1$ for some $n\in \mathbb N$. If $n$ cannot be replaced by a smaller natural number, then $z$ is called *primitive* $n$-th root of unity. There are $\varphi(n)$ primitive $n$-th roots of unity and they are roots of the $n$-th *cyclotomic polynomial* (which has degree $\varphi(n)$). The $n$-th roots of unity can be written as $e^{ \frac{2k\pi}n\cdot i}$ with $0\le k\lt n$.

An important lemma: if $z$ is an $n$-th root of unity, $$ \sum _{k=0}^{n-1} z^k = \begin{cases} n,& z=1 \\ 0,& z\neq 1\end{cases} $$In particular if $z$ is a primitive $n$-th root the sum is zero, a property commonly used in elementary number theory.

The concept can be extended to other fields than $\mathbb C$. For example, in a finite field with $q$ elements, all non-zero elements are $(q-1)$-th roots of unity.

See also this Wikipedia article.