the meaning of "type", "type system", "calculus"

[I expect a number of us will have opinions about this.  Please reply
directly to the poster -- Joe, can you summarize responses?  --BCP]

We all see the following words and phrases quite a bit:

  "type", "type system", "calculus"

What are the defining characteristics of each of these?

I don't expect to see agreement on the meanings of these words and
phrases.  I have asked a number of people already and have never
gotten the same answer twice.  The answers are often not even close.

Joe Wells