If $A$ is a square matrix and $v \ne 0$ is a vector for which
$$Av = \lambda v$$
for some scalar $\lambda$, we say that $v$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda$.
More generally, if $V$ is a vector space over a field $K$ and $T : V \to V$ is a linear transformation, then a non-zero $v \in V$ is an eigenvector of $T$ if there exists a $\lambda \in K$ for which
$$Tv = \lambda v$$