Here's an example. Consider $n = \sum_i n_i$ identical fermions, meaning not more than one particle can occupy a state. Lets say they are hydrogen atom electrons with the discrete hydrogen atom energy spectrum $E_m = - 1/m^2$. Lets say $n_i$ particles are in the energy level $E_i$. Thus the total energy is $E = \sum_i n_i E_i$.
We can treat the particles with energy $E_i$ as a subsystem of the total system. The $i$'th energy level has degeneracy $g_i = 2 i^2$, meaning there are $2i^2$ wave functions with the same energy $E_i = - 1/i^2$. In other words, the $n_i$ particles with energy $E_i$ can take any of the $g_i = 2 i^2 $ states with that energy, but each of the $n_i$ particles can only occupy one state each, we can't have more than one in the same state. This means that we must have $n_i \leq g_i$.
The question is, how many way can we put $n_i$ particles into $g_i$ states (think of them as boxes) such that each box has at most one particle, and further the order in which we put them in doesn't matter. The first particle has $g_i$ possible choices, the second particle has $g_i - 1$ possible boxes, the third has $g_i - 2$ etc... so that we can have $g_i (g_i - 1) ... (g_i - n_i + 1) = \frac{g_i}{n_i!}$ ways to put $n_i$ particles into $g_i$ boxes.
But this assumes the order matters, we overcounted by $n_i!$ permutations. In other words, we get $W_i = {g_i \choose n_i} = \frac{g_i!}{n_i! (g_i - n_i)!}$ total ways of putting $n_i$ particles into $g_i$ boxes (states) where the particles are identical and the order we put them in doesn't matter. This is a measure of the 'disorder' of the system, it quantifies how many different ways can produce the same macroscopic state with energy $n_i E_i$, the total energy of the $i$'th subsystem.
Further when we include another subsystem, say the $j$'th subsystem, we get $W_i W_j = \frac{g_i!}{n_i! (g_i - n_i)!} \frac{g_j!}{n_j! (g_j - n_j)!}$ as a measure of the disorder, i.e. it's a multiplicative measure of the disorder of the system (like probability is multiplicative).
The entropy is an additive 'measure' of this, since things like energy etc... are additive over independent subsystems. To turn a multiplicative quantity into an additive quantity we can take a logarithm, hence $S = \ln(W_i W_j) = \ln(W_i) + \ln(W_j) = S_i + S_j$.
It's sometimes written as $S = k \ln W$ where $W = \Pi_i W_i$ for the subsystems with $W_i$'s and $k$ is a constant used to change degrees to kelvins and vice versa when we define temperature in terms off entropy
@RyanUnger if you want to try make sense of the above, $E_n = - 1/n^2$ is the discrete hydrogen atom spectrum, and $2n^2$ is the $2n^2$ different wave functions with that energy (i.e. diffferent wave functions due tot different values of $l,m,s$ in $(n,l,m,s)$ for $s$ spin, $l$ total angular momentum, $m$ the $z$ component etc...) so the states with energy $E_n$ are the different wave functions
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