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A friend of mine who just finished an introductory calculus class saw my books on fluid mechanics and asked me what a different PDEs look like. I explained elliptical and parabolic by giving examples of a two way and one way space coordinates and time coordinates.
How do I explain hyperbolic without involving too much detail about characteristic? I tried explaining it from the standpoint of the wave equation, but he wanted something a little more mathematical.
Sorry this answer looks so long and scary! It's actually very short and pretty, once you understand the stuff about polynomials. If you're intimidated by the giant mass of text, just skip to the end...
Before I explain what a hyperbolic PDE is, I have to tell you a little about polynomials.
You're probably familiar with the idea of plugging a real number, like $3$, into a polynomial, like $Q(X) = X^2 + 2$. You take a power of a real number by multiplying it by itself a bunch of times, and the "zeroth power" of a real number is just the number $1$, so
$$Q(3) = 3 \times 3 + 2 \times 1 \\ = 9 + 2 \\ = 11.$$
You may know also that you can plug a matrix, like $\left(\begin{array}{cc} 2 & 1 \\ 0 & 1 \end{array}\right)$, into a polynomial. You take a power of a matrix by multiplying it by itself a bunch of times, and the "zeroth power" of a matrix is just the identity matrix, so
$$Q\left(\begin{array}{cc} 2 & 1 \\ 0 & 1 \end{array}\right) = \left(\begin{array}{cc} 2 & 1 \\ 0 & 1 \end{array}\right)\left(\begin{array}{cc} 2 & 1 \\ 0 & 1 \end{array}\right) + 2\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right) \\ = \left(\begin{array}{cc} 4 & 3 \\ 0 & 1 \end{array}\right) + \left(\begin{array}{cc} 2 & 0 \\ 0 & 2 \end{array}\right) \\ = \left(\begin{array}{cc} 6 & 3 \\ 0 & 3 \end{array}\right).$$
An 2-by-2 matrix is just a fancy way of writing down a linear function from $\mathbb{R}^2$ to $\mathbb{R}^2$. For example, the matrix we used before represents the function $A(x, y) = (2x + y, y)$. Multiplying two matrices is the same as composing the associated functions, and the identity matrix represents the function $I(x, y) = (x, y)$. So, if we can plug a matrix into a polynomial, we can also plug a linear function into a polynomial:
$$Q(A) = A \circ A + 2I$$
$$Q(A)(x, y) = A(A(x, y)) + 2I(x, y) \\ = A(2x + y, y) + 2(x, y) \\ = (4x + 3y, y) + (2x, 2y) \\ = (6x + 3y, 3y).$$
Let's call the set of differentiable functions $\mathcal{F}$. You can think of differentiation as a linear function from $\mathcal{F}$ to $\mathcal{F}$. Since we know how to plug a linear function into a polynomial, we can plug the operation of differentiation into a polynomial!
$$Q(\tfrac{d}{dx}) = \tfrac{d}{dx} \circ \tfrac{d}{dx} + 2I$$
$$Q(\tfrac{d}{dx})\sin(x) = \tfrac{d}{dx}[\tfrac{d}{dx}\sin(x)] + 2\sin(x) \\ = \tfrac{d}{dx}\cos(x) + 2\sin(x) \\ = -\sin(x) + 2\sin(x) \\ = \sin(x).$$
Hold that thought.
Say you have a second-order polynomial in two variables,
$$P(X, Y) = \alpha_{20} X^2 + \alpha_{11} XY + \alpha_{02} Y^2 + \alpha_{10} X + \alpha_{01} Y + \alpha_{00}.$$
If you plot all the pairs of real numbers $x$ and $y$ for which $P(x, y) = 0$, you'll generally get a smooth curve---an ellipse, a parabola, or a hyperbola. It makes sense to call a polynomial "elliptic" if its zero-set is an ellipse, "parabolic" if its zero-set is a parabola, and "hyperbolic" if its zero-set is a hyperbola.
If you plug the operations $\tfrac{d}{dx}$ and $\tfrac{d}{dy}$ into $P$, you get a really simple way of writing down a differential equation...
$$P(\tfrac{d}{dx}, \tfrac{d}{dy})f = 0$$
...that would look really complicated in more familiar notation.
$$\alpha_{20}\tfrac{d^2f}{dx^2} + \alpha_{11} \tfrac{d^2f}{dx\,dy} + \alpha_{02} \tfrac{d^2f}{dy^2} + \alpha_{10} \tfrac{df}{dx} + \alpha_{01} \tfrac{df}{dy} + \alpha_{00} f = 0.$$
You might be tempted to call this differential equation "elliptic" if $P$ is an elliptic polynomial, "parabolic" if $P$ is a parabolic polynomial, and "hyperbolic" if $P$ is a hyperbolic polynomial. This actually turns out to be a really good way of classifying differential equations, so the names stuck!
Another quite neat, more physics oriented intuition can be given by comparing two classic examples of parabolic and hyperbolic equations. Consider for instance the "prototype" for parabolic equations, the heat equation $$\partial_t T = \partial_{xx} T.$$ The diffusive term on the RHS will drive the initial distribution $T(x)$ to equilibrium. Now suppose you start with a Gaussian initial profile and enforce no-flux boundary conditions $\partial_x T \cdot n = 0$ (adiabatic system, conservation of energy). What will happen is that the profile will flatten out in time, in particular the peak temperature will decrease. Copy this scenario and let time progress in both setups. Now perform the thought experiment of stopping time and increasing the temperature again in one of the experiments, while leaving the other untouched. Then let the process continue ("unpause time"). What you will observe is that (no matter how large your domain is) the system with the increased temperature will have at every point a different (higher) temperature than the other system! Depending on increase in temperature and domain size this difference might be tiny, but it is there!
Now consider in contrast the simplest hyperbolic equation: The advection equation / linear transport equation: $$\partial_t c + a \partial_x c = 0$$ where $a$ is a constant not depending on $x, t$. It is well-known that this is equivalent to transporting the initial, say, concentration $c_0$ of a substance through space with velocity $a$. Now consider the initial condition $$c_0(x) = \begin{cases} 1 & x \leq 0 \\ 0 & x > 0 \end{cases}$$ and observe the system. What will happen is that the concentration moves for $a>0$ to the right, with velocity $a$. Now again, stop time and increase in one of your experiments the concentration by factor of, say, $2$. "Unpause" time and observe what's happening: For points sufficiently far away from the wavefront (the discontinuity in concentration), nothing has changed! They are still zero, and only notice that something has changed when the wave actually reaches them and sets their value to $2$. This is in stark contrast to the parabolic PDE, where immediately the whole system noticed a difference. Thus, hyperbolic systems exhibit finite speed of propagation (of information). In contrast, for the parabolic heat equation, this speed was infinite!
As a side note: We know that the speed of every process in this universe is limited by the speed of light. That's also why the heat equation is strictly speaking not a correct model. Thus, a relativistic heat equation has been introduced, which is, now not to your surprise, a hyperbolic equation! But since in most practical cases the speed of light is magnitudes higher than the time scales involved in typical problems of heat conduction, the simple standard equation is used assuming infinite speed of information propagation.
The archetypical example, where the term originated from, is the differential equation $\left(aX^2 + bXY + cY^2 + dX + eY + f\right)U = 0$, where $U$ is a function $U\left(x,y\right)$ of $\left(x,y\right)$ and $X,Y$ respectively denote $∂/∂x,∂/∂y$. The classification of equations coincides with the classification of conics given by the corresponding equations $aX^2 + bXY + cY^2 + dX + eY + f = 0$. In that classification, provided $\left(a,b,c\right) ≠ \left(0,0,0\right)$, the curves are either hyperbolas, parabolas or ellipses (for suitable values of $f$). The classification is determined primarily from coefficients $\left(a,b,c\right)$. If they are, themselves, functions of $\left(x,y\right)$, then the character of the equation may change from point to point, and assuming the functions are continuous, then the space of all $\left(x,y\right)$ will divide into contiguous regions or zones, where the equation has the same nature within any given zone.
For example, the equations of the form $a ∂U/∂t + bt ∂^2U/∂t^2 - \left(∂^2/∂x^2 + ∂^2/∂y^2 + ∂^2/∂z^2\right) U = 0$, where $a,b$ are constants and $a, b > 0$ are hyperbolic for $t > 0$, parabolic for $t = 0$ and elliptical for $t < 0$. That includes the Laplace-Beltrami equations for radiation-dominant flat-space varieties of the Big Bang model; i.e. those with line elements of the form $dt^2 - k t \left(dx^2 + dy^2 + dz^2\right)$. (You'll see some really interesting things, when you solve these equations.)
The definition applies much more broadly than this. First, the differential operator need not be linear. Second, there may be more than two coordinates. In that case, the nature of the equation $D\left(X,Y,⋯\right)U = 0$, where the operator $D$ is a function of the second order in the differential operators $\left(X,Y,⋯\right)$, is determined by the Hessian (i.e. the matrix of second derivatives) of the function $D$ with respect to $\left(X,Y,⋯\right)$. In the above example, the signature of the Hessian changes from $\left(+,-,-,-\right)$ for $t > 0$ to $\left(0,-,-,-\right)$ at $t = 0$ to $\left(-,-,-,-\right)$ for $t < 0$. I believe the general classification allows only for signatures where all the signs but one agree. I don't know how an equation whose Hessian has signature $\left(+,+,-,-\right)$ or $\left(+,0,-,-\right)$ would be classified. Those include the equations that would arise in anti de Sitter and Newton-Hooke geometries.
Third, the differential equation may be of the third or higher order. You can find the general classification on the Wikipedia in the article under hyperbolic partial differential equations.
