What does it mean to be moral or amoral? I have heard "Atheists are
amoral!" on one side and "Atheists can be moral without religion" on
the other, but what does that mean?

If being moral means being unselfish, then that doesn't make any
sense. No human action is purposely unselfish, whether it is due to
(perceived) reward of afterlife (or punishment), Imprinted guilt,
'Do-gooder's high', Laws, or some other reward/punishment, perceived
or otherwise.

It seems to me, any action, as long as it benefits me in the long
run is good, taking all things into account (even psychological,
reputation, repercussions, laws, etc.) My foresight may be in error,
leading me in the opposite direction, but the intent is "selfish", I
can't see it being otherwise.

But this is true even to religious folk; Fear of hell, group
pressure, and guilt. All actions they take are meant to directly
benefit themselves, (heaven, acceptance, no guilt, etc.) even if the
presumptions are false. The intent is the same.

So then what are morals? Is it what benefits 'Society'? I don't
think I have reason to benefit 'society' unless that action somehow
benefits me, even in the long run. It would be waste otherwise.

Unless I am missing something the words moral/immoral/amoral seem
meaningless.

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A quote from Albert Einstein on the subject:

"A man's ethical behavior should be based effectually on sympathy,
education, and social ties; no religious basis is necessary. Man would
indeed be in a poor way if he had to be restrained by fear of
punishment and hope of reward after death."