the characteristic families
Some directions in a problem are special — the grain in a plank of wood, the lanes on a highway, the path a ripple actually follows. For a second-order PDE, the characteristic curves are exactly these special tracks: curves along which the equation simplifies, along which signals travel, and across which solutions are allowed to have kinks. Counting how many real ones there are is the same as naming the type.
For A u_xx + 2B u_xy + C u_yy + (lower order) = 0, the characteristic curves are the solutions of the characteristic ODE A (dy)^2 - 2B (dx)(dy) + C (dx)^2 = 0, which factors into dy/dx = (B plus or minus the square root of (B^2 - AC)) / A. The square root of the discriminant decides how many real families exist: hyperbolic (B^2 - AC > 0) gives TWO distinct real families (real, unequal slopes — for the wave equation, x - ct = const and x + ct = const); parabolic (B^2 - AC = 0) gives ONE repeated real family (the two slopes coincide — for the heat equation, the lines t = const); elliptic (B^2 - AC < 0) gives NO real families (the slopes are complex conjugates, so there are no real characteristics at all). Two, one, none — that count is the type.
Characteristics are not bookkeeping; they are the physical skeleton of the solution. Along a characteristic, a hyperbolic equation reduces to an ODE, which is precisely how the method of characteristics and d'Alembert's formula work. Across a characteristic, the second derivative of a solution is permitted to jump, so characteristics are the curves along which singularities and wavefronts propagate. And because the elliptic case has none, elliptic solutions have no preferred direction along which roughness can travel — which is exactly why they come out so smooth. The whole feel of an equation is encoded in how many characteristic families it carries.
For the wave equation u_tt - c^2 u_xx = 0, the characteristic ODE gives slopes dx/dt = plus or minus c, so the two characteristic families are the straight lines x - ct = const and x + ct = const — exactly the tracks of the right-moving and left-moving waves in d'Alembert's solution.
Count the real characteristic families and you have read off the type: two = hyperbolic, one = parabolic, none = elliptic.
These are the second-order analogue of the characteristic curves of a first-order PDE, but here they come in families counted by the discriminant; the dramatic fact is that elliptic equations have NO real characteristics, which is the root of their extreme smoothness.