a travelling-wave solution
Some solutions of a PDE keep one fixed shape and simply glide across space at a steady speed, like a ripple moving outward on a pond while each humped ridge holds its profile. A travelling-wave solution is exactly that: a solution that depends on space and time only through the single combined variable x - c t (or x + c t for the other direction), so it is a frozen profile sliding along at speed c.
Concretely, you look for a solution of the form u(x,t) = f(x - c t). The clever part is what this does to derivatives: u_t = -c f'(x - c t) and u_x = f'(x - c t), so the PDE collapses into an ordinary differential equation for the single-variable profile f. For the transport equation u_t + c u_x = 0 it gives -c f' + c f' = 0, true for every shape f — every profile travels. For a richer equation it gives a real ODE for f whose solutions are the allowed wave shapes; the wave's speed c is often not free but pinned down by requiring a sensible profile.
Travelling waves are one of the deepest organizing ideas in PDEs. They are the natural solutions of the wave equation (left- and right-movers), they appear as fronts in reaction-diffusion equations (a flame or a population invading at a fixed speed), and as solitons in equations like Korteweg-de Vries that keep their shape against all odds. Reducing a PDE to an ODE for f is also a standard first move when you want to understand a hard nonlinear equation.
For Burgers' equation with viscosity, u_t + u u_x = nu u_xx, looking for u = f(x - c t) gives an ODE whose solution is a smooth step (a tanh shape) joining a high value on the left to a low value on the right — a moving, smeared shock front of fixed profile and speed c.
The ansatz u = f(x - c t) turns a PDE into an ODE; the surviving profiles are the travelling waves.
Travelling-wave form is an ansatz (an educated guess), not a guarantee: many PDEs have travelling waves only for special speeds c, and many solutions are not of this form at all. Finding one shows what is possible, not what every solution does.