Let $E$ be an elliptic curve over a field $k$. Let $\text{End}_k(E)$ denote the endomorphism ring of $E$, i.e., $$\text{End}_k(E) = \{\text{base point preserving morphism} \ f:E \to E\}.$$
Since multiplication by an integer $n$ is an endomorphism of $E$, we have an embedding $$ \mathbb{Z} \hookrightarrow \text{End}_k(E).$$
If this is a proper embedding, then we say the elliptic curve $E$ has complex multiplication.
Why is it called complex multiplication? Does it have anything to do with multiplication by a complex number?