What does the Fed's dot plot actually show?

Four times a year the Fed publishes anonymous projections of where each member thinks rates are heading — the "dots". The market reads the median dot as the Fed's plan, though it's really a collection of individual guesses.

Beginners: what did you think the dots meant before you learned? Experienced members: how much weight do you give the median versus the spread of dots?

The dot plot guide decodes it.