Conceptual explanation of the Single particle partition function The Single particle partition function is defined mathematically as
$$\text{Z=$\sum $}g_ie^{\left(\frac{-E_i}{K_BT}\right)}$$
But what is the physical interpretation of the partition function and it's significance to Thermodynamics? I'm seeking a simple yet understandable intuition.
 A: Partition functions are a measure of the allowed volume in (microscopic-)configuration space for the system, and as such they are the normalizing function for probabilities expressed as volumes in configuration space (and assuming the applicability of the ergodic hypothesis).

I know that this is very abstract, but it is also very general.
A: 
But what is the physical interpretation of the partition function and
  it's significance to Thermodynamics? I'm seeking a simple yet
  understandable intuition.

The partition function has one simple physical interpretation in terms of Thermodynamic functions: Its natural log is proportional to the Free Energy (the proportionality constant is the negative inverse temperature).
The Free Energy, which is well-known from Thermodynamics, is given by
$$
F=E-TS\;,
$$ 
where E is the Thermodynamics Energy, T is the temperature, and S is the Entropy.
From a Statistical Mechanics perspective, remember that the probability to be in a state $n$ is given by
$$
p_n\equiv \frac{e^{-E_n/T}}{Z}\;,
$$
where $Z$ is your partition function, T is the temperature, and $E_n$ is the energy of state $n$.
The statistical definition of Entropy is
$$
S=-\sum_n p_n \log(p_n)
$$
$$
=-\sum_n p_n (-E_n/T-\log(Z))
$$
$$
=E/T+\log(Z)
$$
I.e.,
$$
S-E/T=\log(Z)=-F/T
$$
