[Sniffnoy]

You can think of c as the conversion rate between space and time; then, light (and anything else without mass, such as gravity or gluons) travels at a speed of 1. Everything else travels at a speed of less than 1.

(Physicists will in fact use the c=1 convention when keeping track of the distinction between distance units and time units is not important. A related convention is hbar=1.)

You can tell that c is fundamental, rather than just a property of light, from how it appears in the equations for Lorentz boosts (length contraction and time dilation).