Minimal models are 2d conformal field theories whose spectra consist of a finite number of representations of the Virasoro algebra.
The requirement for of a finite number of representations is quite constraining; generically the OPE $ \mathcal{O}_1 \mathcal{O}_2 = \sum_i C_{12i} \mathcal{O}_i$ will contain an infinite number of primaries. In some cases, the right-hand side contains a finite number of operators because the $C_{12i}$ vanish for all but a finite number of $i$. This happens, for instance, in the Narain theories, which enjoy a significantly enlarged symmetry algebra. The other possibility is that the fusion rules for $\mathcal{O}_1$ and $\mathcal{O}_2$ only allow for a finite number of representations on the RHS. The latter option is the one which leads to the minimal models.
In fact, such spectra consisting of an infinite but countable number of degenerate representations do make up consistent theories, which are called Generalized Minimal Models. However, to construct models with a finite number of representations, we will need to be more restrictive.
The fusion product of two representations will only be a finite sum for degenerate representations, so we will restrict to considering these. Recall the definition of a degenerate representation of the Virasoro algebra:
$$ \mathcal{R}_{\langle r, s \rangle} = \frac{\mathcal{V}_{\Delta_{\langle r, s \rangle}}}{U(\mathcal{V}^+) | \chi_{\langle r, s \rangle} \rangle} \, .$$
where $\Delta_{\langle r, s \rangle} = \frac{1}{24} ( c - 1 - 6 ( r b + s / b)^2 ) $, $ b = \sqrt{\frac{c - 1}{24}} + \sqrt{\frac{c - 25}{24}}$, and $| \chi_{\langle r, s \rangle} \rangle$ is a null vector of $\mathcal{V}_{\Delta_{\langle r, s \rangle}}$.
The fusion rules for degenerate representations take the form
$$ \mathcal{R}_{\langle r_1, s_1 \rangle} \times \mathcal{R}_{\langle r_2, s_2 \rangle} = \sum_{r_3 = |r_1 - r_2| + 1}^{r_1 + r_2 - 1} \sum_{s_3 = |s_1 - s_2| + 1}^{s_1 + s_2 - 1} \mathcal{R}_{\langle r_3, s_3 \rangle} $$
where the sums involved run in increments of two. A finite set of such ``singly degenerate“ representations will not be closed under the fusion rules; repeated application of the fusion rules will bring in higher and higher values of $r_3, s_3$, and ultimately an infinite set will be required.
The key to constructing the minimal models will be to start with doubly degenerate representations, defined by
$$ \mathcal{R}_{\langle r, s \rangle} = \frac{\mathcal{V}_{\Delta_{\langle r, s \rangle}}}{U(\mathcal{V}^+) | \chi_{\langle r, s \rangle} \rangle + U(\mathcal{V}^+) | \chi_{\langle r', s' \rangle} \rangle} \, .$$
These are degenerate representations defined by modding out by two null vectors. First of all, it is clear that for this to be consistent, $| \chi_{\langle r, s \rangle} \rangle$ and $| \chi_{\langle r', s' \rangle} \rangle$ must both be null vectors of $\mathcal{V}_{\Delta_{\langle r, s \rangle}}$. This requires that
$$ \Delta_{\langle r, s \rangle} = \Delta_{\langle r', s' \rangle} \, .$$
From our definition above of $\Delta_{\langle r, s \rangle} $, we see that this requires
$$ r b + s / b = -(r' b + s' / b) \, . $$
(Actually a plus, instead of a minus sign is also allowed by the equation, but this is ruled out by the requirement that $p$ and $q$ must be coprime).
Defining $p = r + r'$ and $q = s + s'$, we find $b = -q/p$. Because $r, r', s, s' \geq 1$, we must have $p, q \geq 2$. Furthermore, we require that $p$ and $q$ are coprime. (I think that this is required so that $| \chi_{\langle r', s' \rangle} \rangle$ is a null vector at all, but I'm not sure of the derivation).
Relating $b$ back to $c$ gives the famous formula for the central charge of the minimal models:
$$ c = 1 - 6 \frac{(p-q)^2}{p q} \, , \qquad p, q \geq 2, (p, q) = 1 . $$
Next we should check that this construction does indeed yield a finite set of reps which are closed under fusion.
The fusion rules for doubly degenerate representations are more restrictive than those of their singly degenerate counterparts. These take the form
$$ \mathcal{R}_{\langle r_1, s_1 \rangle} \times \mathcal{R}_{\langle r_2, s_2 \rangle} = \sum_{r_3 = |r_1 - r_2| + 1}^{\min(r_1 + r_2,2 p - r_1 - r_2) - 1} \sum_{s_3 = |s_1 - s_2| + 1}^{\min(s_1 + s_2, 2 q - s1 - s2) - 1} \mathcal{R}_{\langle r_3, s_3 \rangle} \, .$$
We see that both $r_1 + r_2$, and $r_1' + r_2'$ are now given constraints on upper range for $r_3$ (and the same for $s$).
Now, for a given central charge, which defines $p$ and $q$, there is an upper bound in the possible values of $r_3$ and $s_3. So only a finite number of representations can possibly be generated by fusion. At a given central charge, the set allowed representations are called Kac table. See below for an example.
The minimal models we've just constructed are only unitarity when $q = p+ 1$. Non-unitarity models are consistent but cannot be given a quantum-mechanical interpretation. Nonetheless, they are interesting objects to study and often have applications in statistical systems.
(I'll have to give a thorough discussion of unitarity later)
Consider $c = 1 / 2$, which implies $p = 4, \, q = 3$. The Kac table of doubly degenerate representations is given by
$$ \begin{array}{c}\begin{array}{c|ccc} & r = 1 & r = 2 & r = 3 \\ \hline s = 1 & \Delta = 0 & \Delta = \frac{1}{16} & \Delta = \frac{1}{2} \\ s = 2 & \Delta = \frac{1}{2} & \Delta = \frac{1}{16} & \Delta = 0 \end{array} \end{array} $$
This example is the 2d critical Ising model, as we will see in more detail below. There are only three unique representations, as e.g. $\Delta_{\langle 1, 1\rangle} = \Delta_{\langle 3,2 \rangle}$.
Primaries in degenerate representations have vanishing descendents, by definition. The $\sigma$ operator in the Ising model is the field corresponding to the doubly degenerate representation
$$\mathcal{R} = \mathcal{R}_{2,1} = \mathcal{R}_{2,2} \, .$$
Let's determine the level-two descendent which vanishes. This is the same as the primary of the subrepresentation we mod out by, defined by
$$ | \chi_{2,1} \rangle = L_{2,1} | \Delta_{2,1} \rangle = (a L_{-1}^2 + L_{-2} ) | \Delta_{2,1} \rangle \, .$$
This will be a primary if $L_1 | \chi_{2,1} \rangle = 0$ and if $L_2 | \chi_{2,1} \rangle = 0$. Using the $c = 1/2$ Virasoro algebra, and solving these constraints, we find
$$ L_{2,1} = -\frac{4}{3} L_{-1}^2 + L_{-2} \, .$$
Therefore the degenerate field $\sigma$ must satisfy
$$ L_{2,1} \sigma(z) = 0 \, .$$
This equation will allow us to compute the four-point function
(To Be Finished Later)
Like most of the 2d CFT content, this page follows the excellent notes of Sylvain Ribault. The most complete form of those notes is given here
The derivation of the 2d Ising model correlation function using the BPZ equation is given in Yin's lecture notes