The four black pieces on the left are Black Jade. The blue pieces on the right are all high quality Lapis from Afghanistan.

The two pieces on the far left are plain blue Sleeping Beauty Turquoise.
The three pieces at the top are Chinese Turquoise. 
The two pieces at the far right are a jade-green color of Turquoise.
The three pieces in the center are blue, green, and blue-green Chrysocolla.

These are various color shades of Sugilite. Sugilite varies greatly in color from light to dark and various textures from even to heavily mottled.

The two pieces on the far left are Pristine. Pristine is a pure white material. Some think it looks like porcelain.
The three pieces on the upper-right are Bamboo Coral. Bamboo Coral looks very much like Ivory and is an excellent substitute for Ivory.
The two large round white cabs at the bottom are white Coral.

The two pieces on the left are Meteorite.
The piece on the right is Dinosaur Bone.