Precious gemstones found throughout Skyrim. Based on value, from the top in clockwise order are: Diamond, Emerald, Sapphire, Ruby, Amethyst and Garnet.
Giants and dragons can sometimes drop precious gems.
With the acquisition of the Prowlers Profit ability, Precious Gems of all sorts and all qualities will appear in a large number of places, 100% of the time. The Prowler's Profit ability spawns Gems of varying qualities in any container permanently, from only one gem to rarely up to 10 gems in Boss Chests and Dragons.
Giants will always carry gems, mostly Sapphires and Emeralds it seems.
Dragons will always carry gems, always carrying at least one of Flawless Quality. Legendary Dragons (Dragonborn) will carry the highest chance of dropping Flawless Diamonds.
Burial Urns will supply a lot of these gems, along with a small amount of Gold. Found in Nordic Ruins and various tombs and dungeons, Urns are the most common replenishable supply of Gems and small gold.
The corpses of Draugr always have them. Note that these corpses are the ones that DONT attack you. These Draugr are static containers found in varying areas. With the addition of Dragonborn addon, Stalhrim deposits contain Draugr when mined out, and are a good source for both the valuable crafting material and gems.
Sometimes Bandit Leaders will carry Precious Gems, very rarely of Flawless Quality.
Boss Chests, miscellaneous chests, chests found in homes and other areas around Skyrim will always carry at LEAST one gem, regardless of the NPC that owns or guards the chest.
Gems may be found while mining. There is roughly a 10% chance of reciveing a gemstone per piece of extracted ore.
There is a Flawless Sapphire and a Flawless Amethyst in the Temple of Dibella in Markarth. Inside the Inner Sanctum on a shelf.
Amaund Motierre carries a Sapphire, an Emerald, and a Diamond.
There is a Flawless diamond, three flawless garnets and a lockbox on a boat directly north of Reachwater Rock just southwest of Sky Haven Temple. Follow the map to the mine and follow the stream across from the cave downstream a bit and it will be on the right.
The miscellaneous quest received at Merryfair Farm rewards randomly generated gemstones.
A Flawless Ruby can be found in the Emperor's bedchambers on board The Katariah.
A Flawless Diamond can be found in Skuldafn Temple in the room right before the end, along with a Flawless Amethyst and a Flawless Emerald underneath a linen wrap in a bowl on the bottom bookshelf.
A Flawless Diamond along with a Flawless Emerald and a Flawless Garnet can be all found in a pot in Skuldafn before the gate leading outside.
2 Flawless Sapphires can always be found in the Ruins of Bthalft, at the end of the Lost to the Ages quest. They are in the chest, near the Aetherium Forge.
Merchants throughout Skyrim also buy and sell gems; many of the merchants that buy and sell jewelry will also buy and sell gems. General goods merchants can also buy gems.
Aside from various Quests and smithing, Gems are an invaluable source of Gold. At high levels, a handful of gems can wipe out an NPC merchants gold sources very quickly. Their minimal weight makes them indespensable as trading or buyback items.
Precious Gemstones should not be confused with Soul Gems, which serve an entirely different purpose.
Flawless gemstones appear slightly different from normal gemstones. Gemstones have a somewhat irregular surface, while flawless gemstones are symetrical.
Flawless Diamonds have the highest value per weight unit in the entire game, thereby offering the most efficient way to rack up especially since they're fairly easy to come by with the Prowler's Profit ability. There is a value of 1,000 for 1 unit of Flawless Diamonds. Since The Elder Scrolls V: Dragonborn, this is not true, as the Exquisite Sapphire worth 5000 while weighing 0.2 pounds.