a scalar conservation law
Picture cars on a single-lane road, or heat in a thin wire, or pollutant in a river. Some quantity is neither created nor destroyed inside any stretch — it only flows in at one end and out at the other. A scalar conservation law is the equation that says exactly this for ONE conserved quantity: the rate of change of the amount in any interval equals what flows in minus what flows out.
Turn that bookkeeping into calculus. Let u(x, t) be the density and f(u) the flux — the amount crossing a point per unit time, written as a function of the local density. The total amount in [a, b] is the integral of u, and its rate of change equals the flux in at a minus the flux out at b: d/dt integral_a^b u dx = f(u(a,t)) - f(u(b,t)). Shrinking the interval and using the fundamental theorem of calculus gives the differential form u_t + f(u)_x = 0, i.e. u_t + f'(u) u_x = 0. This is a single first-order quasilinear PDE; the characteristic speed is f'(u), so each level of u travels at its own speed. That single feature — speed depending on height — is what makes the nonlinear theory rich and difficult.
Where it lives: traffic flow (Lighthill-Whitham-Richards), the Burgers equation as the simplest nonlinear model, glacier and sediment transport, chromatography. The crucial honest point is that even with perfectly smooth initial data the solution can develop a discontinuity (a shock) in finite time, because fast parts overtake slow parts. Past that moment the differential equation no longer makes literal sense; one must pass to weak (integral) solutions, and then an extra entropy condition is needed to single out the physically correct one.
Traffic flow: let u be car density and suppose flux f(u) = u (1 - u) (cars move fast when sparse, jam when dense). Then u_t + (1 - 2u) u_x = 0. The characteristic speed 1 - 2u is positive for light traffic and negative for heavy traffic, so a sudden jam (a shock) travels backward against the cars — the familiar phantom traffic wave moving upstream.
One conserved density, one flux: the whole nonlinear story is encoded in the shape of f.
The word scalar means one equation for one unknown; a single conserved scalar law is far better understood than systems of conservation laws (Kruzhkov's theorem gives global existence and uniqueness of entropy solutions in any dimension). For systems, the analogous global theory in several space dimensions is still largely open.