Actually, Neil Armstrong's first words weren't the ones he intended. His historic first words on the moon were supposed to be:

"That's one small step for a man, one giant leap for mankind".
What he actually said was:
That’s one small step for man, one giant leap for mankind”.

Notice the "a" between "for" and "man". Without it, the sentence means basically:

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

Now there is some static on the tapes of the transmission, but there is little doubt of what was said. The blunder was a small and completely understandable one as the pressure must have been immense, but what is shocking is that it has gone unnoticed. Most newspapers at the time printed the exact quote, as do many current encyclopedias. Some current encyclopedias go so far as to give a "corrected" quote, one with the "a" inserted. One of those is Encyclopedia Britannica.

Source: "Big Secrets" by William Poundstone.