A basket setting is a very common gemstone setting used in jewelry. It consists of the stone sitting above one or two wire rings held in place with anywhere from two to eight prongs.
This is how such a setting is made. This is going to be a four-prong setting. First, a small ring made of wire, just slightly smaller than the stone, is measured and soldered into a solid ring. Another ring, slightly smaller than the first, is also cut, formed, and soldered. In each ring four round depressions are filed in, equally spaced apart. Then, a piece of straight wire is bent in the center, and the ends are folded up to perfectly fit two of the depressions in the smaller ring, so it looks like this: <0 with the ends reaching far up over the ring. Another strip of wire is bent in half, and matched up with the other two depressions in the small ring. The wire sections are only being held to the ring by friction, so they are then soldered in place.
The larger ring is then pushed down into place between the four prongs, and soldered into place. The loops in the back are clipped and sanded off, and a rotary tool (such as a dremel) cuts a seat for the gemstone in the wire prongs. The stone is then inserted into its seat and the tips of the prongs are bent over it. The whole thing is then soldered (if the stone is durable enough, otherwise the whole thing is soldered before the stone is set permanently in place) to a ring, earring, bracelet, pendant, or whatever it is supposed to be set in. It is then sanded and polished, and there you have it, a simple, yet strong, stone setting.