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.