What is meant by the vector representation of $\mathrm{SO}(6)$?
I have only encountered the term vector representation in the context of the Lorentz group $\mathrm{SO}(1,3)$, where it refers to the $(1/2,1/2)$ representation, derived from the isomorphism $\mathrm{so}(1,3) \cong \mathrm{su}(2) \oplus \mathrm{su}(2)$ of the Lie algebra.
Unless there is any further context, it's the fundamental 6-dimensional representation: if $$ M^T M=1_6\,,\quad \det M=1\,,\qquad M \;\text{is}\; 6\times 6 $$ so $M$ is an $SO(6)$ matrix, then the representation space is $\mathbb R^6$ and $M$ acts as $$ \mathbb R^6\ni v \to Mv. $$
As already stated in another answer, the vector representation of $SO(6)$ is the fundamental representation. Since you mentioned the $(1/2,1/2)$ representation of $so(1,3)$ I would like to add some examples of non-vector representations.
A simple example of a representation different from the vector representation is the adjoint representation acting on the vector space of $6\times 6$ matrices. Here, $g\in SO(6)$ acts as $A\in M_6(\mathbb{R})\mapsto g^{-1}A g$. Also a spinor does not transform in the vector representation of $SO(1,3)$, but in a projective one, e.g. in $(1/2,0)$ and $(0,1/2)$.
You might check this related question discussing the terminology of the expression "vector representation".
