[This question involves mostly math papers, and may be relevant to graduate students learning to write and cite papers, although this is my only justification for this being a math question.]
Usually papers start out with the title and then the author. Sometimes near the author's name there is the phrase "Communicated by John Doe". Does this mean John Doe told the author the essentials of the paper and then the author wrote down the details? If so, why isn't John Doe just a co-author? Or does this mean something else?
I could not find the answer using Google.