Q:

"That's one small step for man, one giant leap for mankind."

With Neil Armstrong's death today, many news sites are posting articles that quote Neil Armstrong as  "That's one small step for (a) man, one giant leap for mankind.".
My question is, does the quote make sense without the 'a'?
and what is the history of the 'a' or not 'a'? 

A:

"A man" means one man. Without "a", it means man as in the whole population. To me, it doesn't make sense without the "a". "Man" without any articles means the same thing as "mankind".
Originally, there was no article, however. The words were: 

"That's one small step for man, one giant leap for mankind."

However, according to the transcript, "a" was meant to be said. That is the source of confusion. 
This article and this article explain the dropping of "a". Here is the transcript.
Historical accuracy aside, it makes the most sense with "a".

A:

Yes, the 'a' should definitely be there.
As for the history, Armstrong said his radio cut out on the 'a' and  he was misquoted.  NASA's transcript can be found here.

[At the time of the mission, the world heard Neil say "That's one
  small step for man; one giant leap for mankind". As Andrew Chaikin
  details in A Man on the Moon, after the mission, Neil said that he had
  intended to say 'one small step for a man' and believed that he had
  done so. However, he also agreed that the 'a' didn't seem to be
  audible in the recordings. The important point is that the world had
  no problem understanding his meaning.]

A:

Armstrong's quote was originally planned to include the word "a":

That's one small step for a man, one giant leap for mankind.

This makes the most since, since it contrasts the actions of a single man with the consequences for all of humanity.
However, in the excitement of landing on the moon, he miffed the delivery. In the recording, he clearly says That's one small step for man, which ruins the sense of the quote since "man" without the indefinite article would generally be taken as synecdoche for humankind. The quote can be found in both versions in various sources.

