Relation between the $N$ particle partition function and probability? For the 1 particle partition function the probability that the particle is in the state with energy $\varepsilon_i$ is given by:
$$P_i =\frac{e^{-\varepsilon_i \beta}}{Z_1}$$
where $Z_2$ is the 1 particle partition function. In the case of an $N$ particle partition function, if the particles are distinguishable we have:
$$Z=Z_1^N$$
and if they are not we have:
$$Z=\frac{Z_1^N}{N!}$$
But do these relate in anyway to a probability(in the same way that $Z_1$ acts as a normalising factor) and if so the probability of what?
 A: Yes, the Z partition function acts as a normalizing factor to compute the probability that the system of the N particles has an energy $\epsilon_i$. It is obviously more complex in the case of many particles system since a lot of configurations may lead to that energy level. 
For example, consider a 2 particles system with 0 and $\epsilon$ as the allowable energy levels for each particles. Obviously, there is only 1 state (0,0) that leads to the total energy 0 and one state($\epsilon$,$\epsilon$) that leads to the energy level $2\epsilon$. However, there are 2 states leading to the total energy level $\epsilon$ which are (0,$\epsilon$) and ($\epsilon$,0). If both particles are identical we would consider that these 2 states are actually the same which accounts for the $N!$ in the expression you gave. 
To conclude, the Z partition is a way to list all the allowable configurations leading to a macroscopic state. It is the most important quantity in statistical physics since all macroscopic quantities can be computed from its knowledge. Indeed, the entropy(average uncertainty of the system),average energy, Free energy, etc... can all be deduced from the partition function.
