Diamonds are a type of gemstone and one of the most valuable substances on Earth. In Panem, diamonds are associated with District 1, whose primary industry is the manufacture of luxury items (such as jewelry). This makes District 1 very wealthy and creates a favorable relationship between them and the Capitol, whose members are always hungry for the finer things in life. According to Katniss, there is a machine in District 1 that can turn graphite into diamonds. However, this machine may be somewhat useless at the time the series takes place; District 13 used to mine graphite, but since that district was obliterated in the rebellion, graphite may no longer common as a raw material.
For the Tribute Parade of the 74th Hunger Games, the tributes from District 1 (Glimmer and Marvel), were dressed in "tasteful tunics" decorated will jewels. Diamonds may have been among those decorations.
Effie Trinket, the District 12 escort, mistakenly says that putting pressure on coal turns it into pearls. Katniss speculates that Effie meant diamonds, but notes that that is untrue as well (putting pressure and heat on coal can result in diamonds, but this process happens only rarely and can take millennia to complete, so it is unlikely that Katniss would know about this rare outcome). During the 75th Hunger Games, Effie's words became something of an inside joke; Peeta quoted them to Finnick Odair after he found a pearl inside an oyster. Finnick didn't understand, but Katniss burst out laughing.
Cinna used diamonds in some of his designs for Katniss: The dark blue velvet dress she wore in District 2 on her Victory Tour was studded with diamonds - which Johanna Mason described as "gorgeous" - and diamonds were used to decorate one of Katniss' potential wedding dresses.