Why is "C" the speed of light?

We get all sorts of questions in our "Ask a Physicist" inbox, (including a positively disheartening number from people who seem to think it's "Ask a Psychic") but one topic that consistently seems to spark people's imagination and curiosity is the speed of light. What defines it, and why can't anything go faster than that? What happens if we try? Thinking about these questions and trying to find their answers is fascinating and fun in its own right, but more importantly it gives us insight into ...