For definiteness, consider the group $SO(3)$. There is a Lie algebra $so(3)$ given by
$$ [T_a, T_b] = if_{abc}T_c $$
The generators of this algebra can be exponentiated to form the elements of $SO(3)$,
$$ \text{exp}[i\alpha_a T_a] \in SO(3) $$
There are various ways to represent the generators $T_a$ in such a way that they still obey the Lie algebra so(3). This in turn yields different representations of the Lie group, SO(3).
In this context, it seems that the representations we are talking about are specific realizations of the elements of SO(3) -- that is, 3x3 orthogonal matrices with unit determinant. But then the literature also makes statements like "the fundamental representation of SO(3) is given by the 3 component vector v, which transforms under a group element $O \in SO(3)$ via $ v \to Ov$." To my ear they are saying the representation of SO(3) is actually the vector v, and not the matrix O.
So my question is: which is it? Are the representations of a symmetry group the generators of that group, the elements of the group itself, or the objects that the group acts upon?