Gemstones are pieces of mineral used to make jewelry or other adornments. They are also called precious and semi-precious stones. Certain rocks like lapis-lazuli or organic matters like amber aren't minerals but they're used for jewelry making so, they're also considered gemstones. Most gemstones are hard, but some soft minerals are used in jewelry because of their lustre or other physical properties that have aesthetic value.
A gem is its own mineral. Diamonds are formed by coal being heated and compressed for years under unbelievable pressure of the earth for a few years, the carbon under pressure and heat melts into itself forming a diamond.