a characteristic curve
Picture the (x,t) plane on which a first-order PDE lives, and imagine the special highways along which information actually travels. A characteristic curve is one of those highways: a path in the plane along which the PDE simplifies to an ordinary differential equation, so that the solution can be tracked one step at a time as if it were riding the curve.
For the transport equation u_t + c u_x = 0 the characteristics are the straight lines x - c t = constant; on each one the solution u holds a single fixed value, set by the initial data where that line begins. More generally the characteristic through a point is the curve whose direction is dictated by the coefficients of the equation: for a u_x + b u_y = c the characteristics solve dy/dx = b/a, and the unknown u rides along them obeying du/dx = c/a. They are the curves of propagation — the worldlines of a signal carried by the equation.
Characteristics are the X-ray of a first-order PDE. Where they fan apart, data spreads thin; where they bunch together and cross, two different incoming values arrive at the same point and the smooth solution must break (a shock). For nonlinear equations the characteristics depend on the solution itself, so they can bend and collide — which is precisely how shocks form. The same word, with a related meaning, names the curves of fast propagation that classify second-order PDEs.
For Burgers' equation u_t + u u_x = 0, the characteristic carrying value u0 is the straight line x = x0 + u0 t — its slope depends on the data. A tall part of the profile moves faster than a low part, so fast characteristics overtake slow ones and eventually cross.
Characteristics are straight for linear transport but slope-by-data for nonlinear flow — that is why nonlinear waves break.
A characteristic carries the solution but cannot carry independent Cauchy data: prescribing values along a characteristic is either redundant or contradictory. Data must be given on a non-characteristic curve.