Nowadays some of us live in a democracy, and we don't discriminate against half-breeds (of the mineral variety) anymore. We just have gemstones or jewels now, including "stones" that aren't true crystals, for example pearls or amber, rather tricky and unusual crystals like opals, or all kinds of things not found in nature but made by man (e.g. zirconia).
Here I only distinguish between crystalline and non-crystalline jewels. As far as the crystalline stones are concerned, they come in a few basic structures that we will give a first quick look.