A representation of a group $G$ is a pair $(\rho, V)$, where $V$ is a vector space, and $\rho$ is a homomorphism $\rho : G \rightarrow Aut(V)$.
The definition I've been given for a subrepresentation is: a subspace $W$ of $V$ such that $\rho (g)(W) \subset W, \forall g \in G$.
Is the subrepresentation just the subspace $W$, or is it the representation $(\left. \rho \right|_W , W)$?
The subrepresentation is the representation $(\left. \rho \right|_W , W)$. However, it's very common to just refer to this representation as "$W$", much as it is common to refer to a group by its underlying set and not explicitly mention the group operation.