Gems that can be found in Minus21 are called Rare Gems, all the others are normal gems, you can get them by killing NPC's or by spinning at the Keywheel. Gems will give you bonusses if you add them to your weapon:
Basic Gems
Crystal (+2 min dmg) (Cost: 1,500)
Diamond (+4 max dmg) (Cost: 2,500)
Emerald (+6 min dmg) (Cost: 3,500)
Ruby (+6 min dmg, +6 max dmg) (Cost: 4,500)
Opal (+20 min dmg) (Cost: 10,500)
Jade (+40 max dmg) (Cost: 20,500)
Apophylite (+60 min dmg) (Cost: 30,500)
Onyx (+60 min dmg, +60 max dmg) (Cost: 40,500)
Large Gems
Large Crystal (+10 min dmg) (Cost: 10,000)
Large Diamond (+20 max dmg) (Cost: 20,000)
Large Emerald (+20 min dmg) (Cost: 30,000)
Large Ruby (+20 min dmg, +20 max dmg) (Cost: 40,000)
Large Opal (+100 min dmg) (Cost: 100,000)
Large Jade (+200 max dmg) (Cost: 200,000)
Large Apophylite (+200 min dmg) (Cost: 300,000)
Large Onyx (+200 min dmg, +200 max dmg) (Cost: 400,000)
Super Gems
Super Crystal (+15 min dmg, 10% extra exp) (Cost: 50,000)
Super Diamond (+30 max dmg, 10% extra exp) (Cost: 75,000)
Super Emerald (+30 min dmg, 10% extra exp) (Cost: 100,000)
Super Ruby (+30 min dmg, +30 max dmg -10% extra exp) (Cost: 125,000)
Super Opal (+150 min dmg, 50% extra exp) (Cost: 500,000)
Super Jade (+300 max dmg, 50% extra exp) (Cost: 750,000)
Super Apophylite (+300 min dmg, 50% extra exp) (Cost: 1,000,000)
Super Onyx (+300 min dmg, +300 max dmg 20% extra exp) (Cost: 1,250,000)
Master Gems
Master Crystal (+20 min dmg, 25% exp and gold) (Cost: 125,000)
Master Diamond (+50 max dmg, 25% exp and gold) (Cost: 150,000)
Master Emerald (+50 min dmg, 25% exp and gold) (Cost: 175,000)
Master Ruby (+50 min dmg, +50 max dmg, -25% exp and gold) (Cost: 200,000)
Master Opal (+200 min dmg, 100% exp and gold) (Cost: 1,250,000)
Master Jade (+500 max dmg, 100% exp and gold) (Cost: 1,500,000)
Master Apophylite (+500 min dmg, 100% exp and gold) (Cost: 1,750,000)
Master Onyx (+500 min dmg, +500 max dmg, 40% exp and gold) (Cost: 2,000,000)