Hello. Can anyone tell me what Bookrings, bookrays, and bookboxes are?
You can find that in the "terminology" section of the site's Frequently Asked Questions page:
If you're interested in joining one, do read the posts for each specific offer, as people set up different requirements for joining and participating. And you might consider starting a bookring or ray of your own - it's an easy way to get better known to other BCers, and it'll help you see how it works!