Let's say that Club A and Club B allow reciprocals (meaning they have an "reciprocal agreement" or "reciprocal play arrangement" in effect). You belong to Club A, but you'd like to play Club B.
So you go to the Club A golf pro and ask him to arrange a tee time for you at Club B. Club A pro calls Club B pro and asks if the Club A member can come play Club B's golf course. Club B Pro says sure, and sets up the tee time.
Why is this process referred to as "reciprocals"? Because at some point a member of Club B is going to want to play Club A. So Club B's pro will call the Club A pro and say, "Hey, remember when we let your member play here? Now I have a member who wants to play your course, so I need to you to reciprocate."
So reciprocals between private golf clubs boil down to "you let my members play your course, and I'll let your members play my course."
Not all private clubs engage in reciprocal agreements, and at those that do the members should always place their request with their own club pro, who will contact the other club.
Clubs that have reciprocal agreements often make it known to prospective members as a value-added service: Join our club and you'll also be able to play these other clubs.
