Can a for-profit use the word "society" or "association" in it's name?
 What states limit the use of certain names or words in for-profits
and what are the words?  Are there any well known for-profit u.s.
companies with the word "society" in its name.  What about words like
"association", and "council"?

I don't think there's any real prohibition to using any of these words
in a for-profit enterprise...it's just that most companies refrain
from doing so in order to avoid giving a wrong impression.

There are occasional business that break the "rule" however, such as
this one:

http://www.business.com/directory/financial_services/banking/banking_institutions/commercial_banks/society_corp/

Society Corp. in Cleveland.

Most states prohibit the use of "corporation", "incorporation",
"incorporated", or abbreviations such as "inc.", "corp", "llc", etc.
unless you have formally incorporated or formed formal organizations.

However, you can use the other words, "society" (think: Red Hat
Society), Association, etc., unless prohibited by law.

In order to qualify as a non-profit, you have to fill out all those
forms and receive non-profit status from the IRS. The most common
being a non-profit which qualifies under 501(c)(3) of the Internal
Revenue Code.

Even if you form an organization and specifically intend for it to be
a non-profit, in most circumstances, it isn't one until it does
qualify for non-profit status under the Internal Revenue Code. It's
not the name of the organization, but the intent and being granted
that status.

Serenata