Diamonds are a type of gemstone mined by the people of District 1. Because of this, District 1 is one of the wealthiest districts in Panem and is adored by the Capitol. According to Katniss, there is a machine in District 1 that can turn graphite into diamonds. However, this machine is somewhat useless, as District 13 used to mine graphite, and no longer produces it, because District 13 was obliterated in the rebellion.
They were used to decorate Katniss' wedding dresses in Catching Fire. They were also used on tribute's chariots during the opening ceremonies. The dark blue velvet dress Katniss wore to her tour in District 2 was studded with diamonds, and was jealously described by Johanna Mason in Catching Fire.
In the Capitol, diamonds are expensive, but popular. In the districts, except District 1, citizens wouldn't dream of having diamonds, because of their expensive prices.
In The Hunger Games, Effie Trinket, the District 12 escort, mistakenly says that putting pressure on coal turns it into pearls. Katniss speculates on whether she meant diamonds, but notes that's untrue as well. Putting pressure and heat on coal rarely results in diamonds; however, this takes millennia and it is unlikely Katniss would know this. In Catching Fire, Peeta quotes Effie Trinket after he finds a pearl while diving for oysters.