As David says $Q$ is the **unipotent radical** of $P$. The subgroup $T$ is (isomorphic to) the **Weyl group** of the group $G_i\cong GL(V_i/ V_{i+1})$. 

The typical way to realize the Weyl group is as the quotient $N/T$ where $N$ is the normalizer of a maximal split torus $T$ in $G_i$ - this is effectively the same thing as fixing a specific basis of $V_i/V_{i+1}$. The Weyl group rears its head in lots of different ways (most especially as a Coxeter group related to the Dynkin diagram of $G_i$) so this is certainly not the only way to realise it.

As for references, it depends on what kind of approach you want. If you want a treatment of $GL_n$ as an algebraic group then I recommend anything by Carter or Humphreys, or else there is the book by Borel. All of these people work in much greater generality than $GL_n$ though. If you just want to understand $GL_n$, then standard algebra texts like the one of Jacobson might be your best bet. (I have e-copies of some of these. If you want them, email me.)