The usual term is "mast hoops." (Not to be confused with mast bands, which are iron parts permanently secured to the mast in various locations and for various purposes.) In the days of wood ships (and considerably thereafter), mast hoops were made of wood, steamed, bent, and riveted at the joint. (If I remember right, one of the buildings at Mystic Seaport is (or used to be) a mast hoop maker's shop.)
Bluejacket offers laser-cut mast hoops made from thin plywood. Nice, but kind of heavy - especially in smaller scales.
The old, traditional way to make model mast hoops is to start with some thin pine shavings from a plane. Soak them in water, smear them with glue, and wrap them around an appropriately-sized dowel (which you've wrapped first with wax paper). Wrap another piece of wax paper around the result, and hold the whole mess tight with rubber bands. When it's dried into a thin-walled cylinder, slice it into thin bands.
I've had success with paper, sliced from heavy yellow envelopes. (One of them will make enough mast hoops to last you the rest of your life.) A drop of white glue on the end, and a mandrel of the appropriate size, makes the hoop. A touch of wood stain gives it the right color.
The mast hoops have to go over the masthead early in the rigging process - before the crosstrees. I read somewhere that some ships put a few extra hoops on the stack, to avoid the nuisance that would happen if one broke. The sail has grommets sewn in the appropriate locations, in the hem of the sail just inside the boltrope. Lashings hold the grommets to the mast hoops.
When the sail is furled, if the rig has a fixed gaff, the mast hoops stay with the sail; if the sail is removed they fall into a stack at the bottom. If the rig has a hoisting gaff (like both masts on a typical Gloucester schooner), the gaff is lowered to furl the sail, and the mast hoops wind up at the bottom - still secured to the sail. The sail usually falls in such a way that the edge flops into an accordian shape, with the canvas folding alternately to port and starboard. It's a neat effect to reproduce on a model.
Again, I highly recommend the old movie "Captains Courageous." (In fact it gets my vote as my favorite seafaring movie of all those I've seen.) At various points in that movie you can see lots of standard evolutions in working the schooner rig. I watched it for about the 30th time a couple of nights ago; it's worth freezing frames and studying them. It has scarcely anything to do with the Kipling novel; practically nothing survived except the names of the characters (including the ship). To my notion that's not much of a defect; the book isn't one of Kipling's greatest, and the screenwriters' additions actually make the story more interesting. It's worth noting that the story gets updated to the 1930s. The sailors of Kipling's age wouldn't have recognized the schooners in the movie - much less the cars on the streets in various scenes.
The flick is available for a reasonable price on DVD: www.barnesandnoble.com/.../3622889 . Note the cheaper used copies; there probably are others elsewhere on the web.
Hope that helps a little.