A ring is a collection of sites that share a common topic. For example, there are several dog rings. Some of them allow any site that has anything to do with dogs. Several are breed-specific and only allow sites about that breed. Some allow commercial sites like breeders or pet supply stores, while others only permit personal pages dedicated to one's pet.
Each site in a ring has a NavBar – a piece of code that links the site to every other site in the ring. The NavBar has 3 important links: previous, next, and hub.
The "next" and "previous" links exemplify what it means for a site to belong to a ring. Clicking on the "next" or "previous" links takes you from one site to the next/previous site listed in the ring. If you keep clicking through the sites, you'll eventually end up back where you started and will have visited every site in the ring.
The "hub" link takes you to the ring's hub page. This page has a description of the ring and provides more comprehensive access to the ring – you can apply to join the ring, contact the user who manages the ring, see a list of all the sites in the ring, rate and comment on each site, and so on.
Most all rings are managed by WebRing members. For the most part, Ring Managers are autocrats within the rings. They establish the join criteria, decide which applicants are approved for membership in the ring, determine the order of their members' sites, and much more. A good manager keeps the quality of the available content available as high as possible.