a characteristic speed
When a disturbance travels through a medium, it does not move at one universal speed — different 'modes' move at different speeds. Sound in air, the two waves of a vibrating string, the surface and shear waves in an earthquake: each rides at its own pace. A characteristic speed is exactly such a travel speed: the rate at which one particular kind of signal moves through a hyperbolic system.
Mathematically, for a system U_t + A(U) U_x = 0 the characteristic speeds at a state U are the eigenvalues lambda_1(U), ..., lambda_n(U) of the matrix A(U). Each eigenvalue lambda_k comes with a right eigenvector r_k, and the pair (lambda_k, r_k) is the k-th characteristic field: along curves dx/dt = lambda_k(U) in the (x, t) plane — the characteristics — information of that field is carried. For a scalar law u_t + f(u)_x = 0 there is just one speed, the wave speed f'(u): the value u is constant along the straight line moving at speed f'(u). Because f'(u) depends on u, different heights move at different speeds, which is precisely how a smooth profile can steepen and break.
Characteristic speeds are the heartbeat of hyperbolic theory. They are real (that is what 'hyperbolic' guarantees), and the largest and smallest of them bound the domain of dependence: the solution at (x, t) depends only on initial data in the interval [x - lambda_max t, x - lambda_min t]. They fix the maximum signal speed, which in turn fixes the CFL stability limit for numerical schemes, and the gaps between them (strict hyperbolicity) control how cleanly the waves separate.
For Burgers' equation u_t + (u^2/2)_x = 0 the flux is f(u) = u^2/2, so the characteristic speed is f'(u) = u. A bump where u = 2 outruns a region where u = 1: the faster-moving high part catches the slower low part ahead of it, characteristics cross, and a shock forms. The speed literally is the height of the solution.
In a scalar law the characteristic speed f'(u) varies with the solution itself — that is what lets shocks form.
Characteristic speed is not the same as the speed of an individual gas molecule or the fluid velocity; it is the speed of a wave of information. In gas dynamics the three characteristic speeds are u - a, u, and u + a, where u is the flow velocity and a is the sound speed — only the middle one equals the material velocity.