A vector space is a module over a field. Elements of a vector space are called vectors.
Examples.
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Given any field $F$, $\{0\}$ is a vector space over $F$ (the zero vector space).
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Given any field $F$ and $n\in\N^+$, $F^n$ is a vector space with scalar multiplication defined by
\[\lambda\cdot(x_1,\cdots,x_n) = (\lambda x_1,\cdots,\lambda x_n)\]for all $\lambda\in F$ and $(x_1,\cdots,x_n)\in F^n$ (the standard vector space).