Benjamin Dickman already linked to my answer on Math.SE, but I didn't talk too much about exactly the motivation of the characterisation for second order equations. So let me write a little bit about that here.
First, I want to emphasize again that the classification is incomplete: the usual scheme does not include all possible equations.
Let us start by looking at the principal part of the equation (that is the coefficient matrix concerning the second order derivatives) $a^{ij}\partial^2_{ij}$. We know/assume that $a^{ij}$ is a symmetric $n\times n$ matrix. By Sylvester's law, we can classify $a^{ij}$ by its positive, zero, and negative indices of inertia $n_+, n_0,n_-$.
We pause here to note that $n_0$ agrees with the nullity of $a^{ij}$.
I recall here the usual definitions (up to swapping $n_+$ and $n_-$ by an overall sign change):
- elliptic: $n_+ = n$, $n_0 = n_- = 0$
- parabolic: $n_+ = n-1$, $n_0 = 1$, $n_- = 0$.
- hyperbolic: $n_+ = n-1$, $n_0 = 0$, $n_- = 1$.
Note that higher order nullity is already ruled out, and also are the cases where $n_0 = 0$ and $n_- = 2, 3,\ldots, n-2$.
So why do we focus on just these three?
- The parabolic case is a very good model for diffusive phenomenon; this was already noted by Fourier who gave a physical derivation of his heat equation.
- The hyperbolic case is a very good model for wave propagation, and also to some extent dispersive phenomenon; this was noted by d'Alembert who gave a physical derivation of the wave equation.
- The elliptic case is important physically as elliptic equations arise naturally when one considers solutions to parabolic/hyperbolic equations which are stationary in time.
So one motivation of why the literature have largely focused on developing these three types of equations, and not on the other second order PDEs, is that from very early on the physical relevance of these three types of equations are understood.
(I should note here that in the 20th century another second order linear PDE, which by virtue of it having complex valued coefficients does not fit into the existing classification scheme, also rose to prominence, and that is Schrodinger's equation from quantum mechanics. Again, the interest in the equation is driven by physics.)
Why do we classify the equations in these three types?
The answer is simply our human desire to organize knowledge. It turns out that the qualitative features of elliptic, parabolic, and hyperbolic PDEs allow us to write down separate, somewhat comprehensive theories for each of the types. I include some high-lights here:
- For both hyperbolic and elliptic types, but notably not for parabolic type equations (let us assume constant coefficients), we can write down solutions to the Cauchy problem in the case of real analytic initial data using the theorem of Cauchy-Kowalevski.
- For hyperbolic types, this solution to the initial data problem turns out to be "stable", in the sense that for initial data that has finite regularity we can also find solutions of finite regularity. For the elliptic case this stability is lost.
- For elliptic equations, the more natural question to ask is not an initial value problem, but a boundary value problem on a bounded domain $\Omega$. Note that whereas for the initial value problem we usually prescribe both the value of the function and its transversal derivative at the initial time, for the boundary value only one out of the two is prescribed. This is related to the next point.
- For both the elliptic and parabolic types, but notably not for the hyperbolic type equations, we have the notion of maximum principle for solutions.
- Solutions to the elliptic and parabolic type equations enjoy smoothing properties, while solutions to the hyperbolic type equations have propagation of singularities. Examples:
- In the elliptic case: Let $\Omega$ be a suitably regular, bounded domain. Then for any continuous function $f$ specified on $\partial\Omega$ we can solve $\triangle u = 0$ in $\Omega$ with $u = f$ on $\partial\Omega$. Furthermore, we know that strictly in the interior of $\Omega$, $u$ is real analytic (even in the case where $f$ cannot extended to a $C^1$ function).
- In the parabolic case: the heat kernel evaluated with $y = 0$ is formally a solution to the heat equation whose initial value is the Dirac-$\delta$. Note that at all times $t > 0$ the solution is $C^\infty$.
- In other words, solutions to elliptic and parabolic problems are often smoother than their boundary/initial values.
- For hyperbolic equations, singularities in the initial value are propagated into the future, along characteristic curves of the equation. For example, consider the one dimensional wave equation $\partial_t^2 u = \partial_x^2 u$, with initial data
$$ u(0,x) = \begin{cases} x^3 & x \geq 0 \\ 0 & x < 0\end{cases}, \quad u_t(0,x) = \begin{cases} - 3x^2 & x \geq 0 \\ 0 & x < 0 \end{cases}$$
the data here is in $C^2 \times C^1(\mathbb{R})$. The solution to the equation turns out to be
$$ u(t,x) = \begin{cases} (x-t)^3 & x \geq t \\ 0 & x < t\end{cases} $$
and is only twice continuously differentiable, with its third derivative discontinuous along the characteristic line $x = t$.
- Concerning the initial value problem, the parabolic type equations only have existence forward in time, while the hyperbolic type equations have existence both forward and backward in time.
- Somewhat more technically: with elliptic problems it is often very convenient to work in spaces of functions which are Holder continuous, with occasional call to use Lebesgue/Sobolev spaces of various Lebesgue exponents. Similarly for parabolic problems. For hyperbolic equations in spatial dimension $>1$, pretty much the only good tool is via $L^2$-based Sobolev spaces.
- Hyperbolic, and not elliptic or parabolic (at least in the linear case, for nonlinear equations there are some exceptions) enjoy the property of "finite speed of propagation". (In the case of the linear wave equation $\partial_t^2 u = c^2 \triangle u$ this says, in particular, that if the initial data is zero outside of a ball of radius $R$, the solution at time $T$ is zero outside the ball of radius $R + c|T|$.) This is noticeably false for the initial value problem for the heat equation.
Lastly, note that if your equation has constant coefficients, then through a linear change of coordinates you can bring your equation into the form of a wave/heat/Laplace equation if it is classified as hyperbolic/parabolic/elliptic. So you can also motivate the general definition thus as saying that a hyperbolic equation looks, in a small neighbourhood, like the wave equation plus some small perturbations etc. This has the advantage of leading to a discussion of the method of freezing coefficients for elliptic estimates, say.
