On this MIT lecture, the difference between the heat equation and the wave equation includes signal travelling infinitely fast in the heat equation, while it has finite speed in the wave equation:
I guess I don't get the idea of "signal" because in the heat equation there is a partial derivative with respect to time of the function that assigns a temperature to each point in space at each time, and this partial derivative with respect to time, which is not infinite, would seem to be the speed.
So what is it exactly that travels infinitely fast. After a Google search it seems as though this is the "paradox of instantaneous heat propagation" and it gets into other equations, and even has a relativistic counterpart. In other words, I couldn't find any entry-level explanations.