Olivine is a family of minerals that include 2 minerals that range in composition from iron silicate (foresterite) to manganese silicate (fayalite).
Olivine is common in most dark igneous rocks to some degree. When in liquid solution trapped in the gas bubbles of lava, the olivine can cool slowly allowing the gem variety to crystallize as peridot. This limy green gem material is much rarer than the abundant olivine.
Arizona is famous for the peridot that forms in the lavas. Most of it is in the form of compressed grains, that really don’t get big. However, sometimes, pieces large enough to cut and facet are found. Pea-size pieces are good for tumbling. Larger grainy pieces can be sealed and slabbed for displaying. Beads are common because of the color, though most are small.
I just spent a week in Arizona trying to collect peridot. It is ALL on Indian reservations and off limits to collecting, unless you belong to one of the few tribes that control the collecting beds.
Afghanistan has produced some single crystals to 1.5inches, but those are exceptionally rare and always expensive.