Understanding notation regarding particles states and wavefunctions In the development in my notes of second quantisation I have a problem in understanding notation.
We start by considering a basis $\psi_i(\mathbf{r})$ for the Hilbert space of single particle wavefunctions and then build it to a basis of the Hilbert space of two particles (bosons or fermions). We then generalise this to the case of $N$ particles.

For $N$-particle systems the wavefunction can be written as $\psi_{i_1,\dots,i_N}(\mathbf{r}_1,\dots,\mathbf{r}_N)$. We want to build a basis similar to the case of two particles $\dots$ Let $\psi_{i_1,\dots,i_N}(\mathbf{r}_1,\dots,\mathbf{r}_N)$ be the many particle state in which the single particle states $i_1,\dots,i_N$ are occupied (for bosons some or all of the states may coincide). One might at first consider the product state $\psi_{i_1}(\mathbf{r}_1)\dots \psi_{i_N}(\mathbf{r}_N)$ $\dots$

I cannot understand what this means physically. If we know there is a particle(s) in state $i_1$ why do we need $\mathbf{r}_1$? Does the state $i_1$ not specify position? Also what exactly does this mean by state, I thought that the wavefunction was the state? Furthemore how does this relate to a site?
 A: $\newcommand{\ket}[1]{\lvert #1 \rangle} \newcommand{\bra}[1]{\langle #1 \rvert}$The states of a quantum system are nothing else than the abstract vectors in the Hilbert space of states $\mathcal{H}$. For one particle, given a basis of position eigenkets $\ket{x}$ with $\hat{x}\ket{x_0} = x_0\ket{x_0}$ and a state $\ket{i}\in\mathcal{H}$, the wavefunction is
$$ \psi_{\ket{i}}(x) := \langle x \vert i \rangle $$
and since the map $\mathcal{H} \to L^2(x), \ket{i} \mapsto \psi_{\ket{i}}$ is an isomorphism of the abstract space of states to the space of wavefunctions, the wavefunction contains all information the abstract state contains. Thus, you may also say that the wavefunction is the state.
If we have more than one particle, the space of states of the total system is the $N$-fold tensor product of the single particle states, and the wavefunction is then
$$ \psi_{\ket{i}}(x_1,\dots,x_N) := (\bra{x_1}\otimes \dots \otimes \bra{x_N})\ket{i}$$
where $\ket{i}$ is an $N$-particle state that may consist of single-particle states $\ket{i_1},\dots,\ket{i_N}$ as $\ket{i} = \ket{i_1}\otimes\dots\otimes\ket{i_N}$, but could also be an entangled state for which such a decomposition is impossible.
A: 
If we know there is a particle(s) in state $i_1$ why do we need
  $r_1$? Does the state $i_1$ not specify position?

The fact that the particle is in a state $|\psi\rangle$ does not specify the particle's position - it specifies the wave-function
$$
\langle x | \psi \rangle = \psi(x)
$$
from which one can find
$$|\psi(x)|^2$$ 
This is the probability density for observing the particle $\psi$ at position $x$ -  a distribution over $x$. It does not specify that the particle $\psi$ is at the position $x$. 

what exactly does this mean by state, I thought that the wavefunction
  was the state?

The wave-function $\psi(x)$ is one representation of the state - a particular choice of (position) basis states, in this case
$$|\psi\rangle  = \int dx \psi(x) |x\rangle$$
This is similar to representing a wavefunction as vector $\psi_i$ in some basis states $|i\rangle$, e.g.
$$|\psi\rangle  = \sum \psi_i |i\rangle$$
