A rivoli is...

 A rivoli is a crystal with a foil backing, making it very sparkly. There are no holes, so it's a cabochon, not a bead. They come in a variety of shapes and sizes, for many creative possibilities.

 The foil backing needs to be protected from scratches and wearing off, because if that happens, the crystal will lose some sparkle. Before working with the rivoli, paint some clear nail polish on the foil back and let it dry.

I beaded a peyote bezel around my rivoli before adding tatting around it. Deborah Roberti of AroundTheBeadingTable.com has written an excellent free tutorial for creating a peyote bezel, and that's what I followed. She has many more patterns for sale on her website. 

I used a 12mm Swarovski rivoli, size 11/0 Delicas and size 15/0 seed beads. My beading thread was 6 lb Fireline, which is a little thicker than the 4 lb Fireline recommended in the tutorial, but it worked well for me. I must point out that it's important to use a thread especially made for beading to do the peyote, because an ordinary sewing thread can easily break from the stress. 

The necklace I designed. The center is a beaded rivoli, with tatting added around it. I joined the tatting by inserting a tiny crochet hook under the beading thread between beads. Tatting techniques used included bead picots, beads on the core thread, Josephine rings, and split rings. I used size 20 thread in my own hand dyed "Lilac Sea" and "Forest", plus Lizbeth white.

I chose to finish the necklace with strung beads on a flexible beading wire.

Tatting the necklace in progress. I was surprised at the creamy color of the size 11/0 Toho Lt. Jonquil seed beads on the white thread, as compared to the darker color in the tube. I was very pleased with the results!