Dan, I am always impressed with the depth and breadth of your knowledge-- a Constitutional scholar in addition to everything else. You must make one heck of a college student.
For those who don't bore easily, a little more about "freedom of association," if I can recall correctly. I am quite rusty and out of date on this stuff, so don't go to court based on anything I say here.
As Dan notes, the concept of a Constitutional right of free of association was developed (some say invented) in late 1950's and 1960's and applied to strike down southern state laws aimed at hindering the desegregation and anti-discrimination work of the NAACP and later the ACLU. While certainly not explicit anywhere in the Constitution, the right is said to derive from the First Amendment rights of free speech and peaceable assembly (to petition the government to redress grievances.) One needs to be able to freely form political associations with like-minded people in order to effectively assemble and petition, or so the reasoning usually goes.
Not all discrimination by "associations" is protected. For example, the right did not justify business associations' (such as the jaycees and rotary club) discrimination against women. I do not remember the exact test used, but am pretty sure it was a balancing of interests between the need or reason for the exclusion vs. the harm done by the discrimination. I also recall that it is a much more lenient test than what is required when other rights are in question.
I believe that the scope of the protection is still relatively undefined, but I do recall a few general guidelines: Political and activist associations are given a much higher degree of protection than non-political and social groups (Martha's advocacy organization gets more protection than Hootie's club); organizations that are more private (strict rules of admission, very narrow segment of the public admitted) may receive more protection than clubs that are less private (clubs that are generally open to all but one segment of the public); organizations that are actively engaged in or promoting business and commerce receive relatively less protection (this is likely why Augusta goes to great lengths to say that the tournament is entirely separate from the club;) Courts have taken the status of the group being excluded into consideration when balancing the right-- when minorities or women were excluded , the associations received less protection (the cases of this era considered discrimination against minorities suspect, and discrimination against women semi-suspect.)
Some question whether the right should or does provide any protection to groups that are formed for purely social purposes. This looks to be the case, but I don't think the question has been definitively answered. I do recall a few cases where certain social activities, such as choosing a dance partner, were not protected associations (I think one case involved a law which prohibited those above a certain age from dancing with those under a certain age.)
Dan is also correct that the "freedom of association" is considered by many to be a right which is found outside the text of the Constitution-- a penumbra right-- thus creating a problem for anyone who likes the right, but claims to be a strict constructionist. (Shivas, are you a strict constructionist?). Association is much less controversial than its brethren penumbra right -- the right of reproductive privacy (better known either as freedom of choice, or freedom to murder, depending on your point of view) and the foundation of Roe v. Wade. But both rights were recognized and expanded in the same S. Ct. era and rely on similar logic. (Griswold, mentioned above by Dan, and a case named Eisenstadt v. Baird laid the foundation for Roe v. Wade.) [Interesting internal jockeying on Eisenstadt and Roe: The S.Ct. heard argument on Roe v. Wade, put off ruling, then heard and decided Eisenstadt, including a few key phrases that worked perfectly for abortion, then reheard and came down with Roe v. Wade. Like him or not, Wm. Brennan was one smart cookie.)
Some conservative scholars and judges gloss over the extra-textual nature of the right by claiming that they can see it right there in the first amendment (they have better vision than me). Others see it flowing so directly from the first amendment as to not be a penumbra right at all (or hardly).
A few ironies regarding freedom of association as it applies to the Augusta situation. First, the right was created in the fight against discrimination and desegregation in the South. Second, the right which so many uphold as one of the founding principles of our nation and Constitution is but a handful of decades old, and has little more specific textual Constitutional basis than that longtime conservative whipping post: the right to reproductive privacy.