The inner ring is the social circle to which people conform to the center. People are groupish. Their instincts tell them to do things that knead them deeper into groups. They often do this with multiple groups at the same time. Lewis warns that we do things to become part of the inner ring which we wouldn't do otherwise, sometimes immoral things. And it's not like we do it deliberately, Our perception of right and wrong becomes skewed by ingroup/outgroup biases.
What counts as an inner ring? According to Psychologists' Minimal Group Paradigm, pretty much everything. The minimal conditions required for discrimination to occur between groups is in fact, very minimal. Anything can trigger ingroup discrimination, even the most trivial distinctions, like whether you over or under estimated the number of dots on a page. The distinction merely needs to be an object of our attention in order to trigger team based thinking.
So finish the sentence, "I am..." An evangelical? Working class? A liberal? A parent? There's your inner ring. There's your tribe. Now you will do things to protect your tribe from outsiders and prove to your tribe that you really are one of them. We... whatever it is... stick together, even when we're wrong.
"There are no formal admissions or expulsions. People think they are in it after they have in fact been pushed out of it, or before they have been allowed in: this provides great amusement for those who are really inside. It has no fixed name. The only certain rule is that the insiders and outsiders call it by different names. From inside it may be designated, in simple cases, by mere enumeration: it may be called “You and Tony and me.” When it is very secure and comparatively stable in membership it calls itself “we.” When it has to be expanded to meet a particular emergency it calls itself “all the sensible people at this place.” From outside, if you have dispaired of getting into it, you call it “That gang” or “they” or “So-and-so and his set” or “The Caucus” or “The Inner Ring.” If you are a candidate for admission you probably don’t call it anything. To discuss it with the other outsiders would make you feel outside yourself. And to mention talking to the man who is inside, and who may help you if this present conversation goes well, would be madness."