a reaction-diffusion equation
Picture a population of animals spreading across a landscape, or a chemical reacting while it also diffuses through a gel. Two things happen at once: stuff spreads out smoothly (diffusion, the same blurring that flattens heat), and stuff is created or destroyed locally (reaction — births and deaths, or molecules combining). A reaction-diffusion equation puts these two effects into one equation and lets them fight. The tension between them produces almost everything interesting in mathematical biology and chemistry: travelling fronts, spots, stripes, oscillations.
The model equation is u_t = D Laplacian u + f(u). Read it term by term: u(x, t) is the density (of a species, a chemical, a temperature); the left side u_t is the rate of change in time; D Laplacian u is diffusion, which always pushes u toward its local average (D > 0 is the diffusion coefficient); and f(u) is the reaction, a function of the local value of u that adds or removes material. The reaction's fixed points — the values u* where f(u*) = 0 — are the equilibrium states the dynamics organize themselves around, and whether each is stable (f'(u*) < 0) or unstable (f'(u*) > 0) governs which states persist and which fronts invade which. With one species f(u) is scalar; with several interacting chemicals you get a SYSTEM of coupled reaction-diffusion equations, which is where Turing patterns live.
Why is this the workhorse of nonlinear PDE applications? Because it is the simplest setting where diffusion's smoothing competes with a nonlinear source, and that competition is exactly what generates self-organization. The Fisher-KPP equation (f(u) = u(1 - u)) describes an advancing population front; the Allen-Cahn equation (f(u) = u - u^3) describes phase boundaries sharpening; the Belousov-Zhabotinsky reaction makes literal coloured waves in a dish. Crucially, the same diffusion that would just smooth and decay on its own becomes, when paired with the right reaction, the engine of pattern formation — which is counterintuitive and is exactly Turing's insight.
Take f(u) = u(1 - u) (logistic growth) and D = 1: u_t = u_xx + u(1 - u). Start with a small bump of population near x = 0 surrounded by empty space. The bump grows (reaction) and spreads (diffusion), and after a while it organizes into a wave of constant shape moving at speed 2 into the empty region — a travelling front invading u = 0 and leaving u = 1 behind.
Diffusion spreads, reaction grows: together they make a moving front.
Diffusion alone only smooths and flattens; reaction alone is just an ODE at each point. The interesting behaviour — patterns, fronts, blow-up — comes only from the COUPLING, and (for Turing patterns) often needs several species diffusing at DIFFERENT rates.