a first-order hyperbolic system
Imagine several quantities — say the density, momentum, and energy of a gas — that all ride along together and influence one another as they travel. Each carries information at its own finite speed, and the whole bundle is a system of first-order equations. 'Hyperbolic' is the technical word for the case where everything genuinely propagates: there are real, distinct travel speeds, no instantaneous action at a distance, and the future is built from the past along travelling fronts.
Concretely, write U(x, t) = (u_1, ..., u_n) and the system in the form U_t + A(U) U_x = 0, where A is an n-by-n matrix (the flux Jacobian when the system comes from conservation laws U_t + f(U)_x = 0, so A = f'(U)). The system is hyperbolic at a state U if A(U) has n real eigenvalues lambda_1, ..., lambda_n with a full set of real eigenvectors. The eigenvalues are the characteristic speeds — the speeds at which information moves — and each eigenvector picks out a 'wave field' that travels at its speed. Diagonalising A locally turns the coupled system, near a smooth solution, into n nearly independent transport equations, one per characteristic.
Why insist on real eigenvalues and a full eigenbasis? Because that is exactly what makes the Cauchy problem well-posed: data given at t = 0 determines the solution, with finite domain of dependence and continuous dependence on the data. If A had complex eigenvalues the problem would be elliptic-like and ill-posed as an initial-value problem (think of trying to predict Laplace's equation forward in 'time'). Hyperbolic systems are the mathematical home of waves, gas dynamics, shallow water, and electromagnetism in vacuum — wherever signals travel at finite speed.
The 1-D wave equation u_tt = c^2 u_xx becomes a first-order system by setting v = u_t and w = c u_x: then v_t = c w_x and w_t = c v_x, i.e. U_t + A U_x = 0 with U = (v, w) and A = [0, -c; -c, 0]. The eigenvalues of A are +c and -c — the two characteristic speeds, the left- and right-moving waves.
Rewriting the wave equation as a 2-by-2 hyperbolic system exposes its two finite wave speeds.
Hyperbolicity is a property of the matrix A at a given state, so for a nonlinear system A = f'(U) it can hold for some states and fail for others. A system can also be only weakly hyperbolic — real eigenvalues but a missing eigenvector — and then the initial-value problem can lose well-posedness even though the speeds are real.