Each claw is unique. Each one is made of precious gems and/or materials and can be very valuable. Each Dragon claw will open at least one puzzle door of the ancient Nordic dungeon in which it is found. The solution to the puzzle door's combination is found on the underside of its corresponding claw. Zooming in the claw in the Items menu will make the solution easier to see.
These are large, open hallways with a puzzle door at the end. The walls of these hallways tell many stories, as the name suggests. One image seems to depict many Draugr bowing towards an almighty Dragon Priest.
The Coral Dragon Claw can be obtained in two ways.
The first is from Birna, the shop keeper in Winterhold. She claims she purchased it, as she was led to believe that it would lead her to great treasure, but it never panned out for her. She offers it for 50 in exchange for telling her if it ever "works" in the future. (However, if it is returned after being used, there is no option for telling her.) Along with the sale, she reveals a possible location — the Yngol Barrow, located Northeast of Windhelm.
If not obtained from Birna the Coral Dragon Claw can be found at Yngol Barrow, on a pedestal.
The Ebony Dragon Claw is found at Korvanjund during the quest The Jagged Crown. Depending on which side of the war is supported, it is found in the Hall Of Stories near the dead bodies of either two Imperial Soldiers or two Stormcloaks.
Note: Because of its jet-black coloration, it may be difficult to see at first glance, but it is set on a beige floor, so it may not be.
The Emerald Dragon Claw opens the first Door Puzzle in Reachwater Rock. It is found near the entrance, on a pedestal next to an Adventurer's dead body.
The Glass Dragon Claw is found in Forelhost during the Siege on the Dragon Cult quest. It is in the Forelhost Refectory resting on a pedestal in the top room that resembles a library directly before the spear-gate. While taking the Glass Dragon Claw can cause the spear-gate to close, setting any other object on the pedestal can counteract the removal of the Glass Dragon Claw.
The Golden Dragon Claw is found at Bleak Falls Barrow. A group of four bandits and a Dunmer named Arvel the Swift came to Bleak Falls Barrow with the stolen Golden Dragon Claw hoping to find treasure behind the Nordic puzzle door. The robbery went wrong, and Arvel was trapped in a Frostbite Spider's web, leaving the rest of the group waiting at the entrance of the dungeon. Retrieving the Golden Dragon Claw from Arvel and returning it to Lucan completes the quest. Thereafter, the Golden Claw is displayed on the counter of the Riverwood Trader, which, for those inclined, makes it easy to steal. After being stolen the claw can be sold back to the Riverwood Trader and bought back again in order to clear the "stolen" status. Additionally, if Camilla Valerius is married by the individual wanting the claw, it can be taken, instead of stolen.
The Ruby Dragon Claw is found and used during the quest Tending The Flames inside Dead Men's Respite. It is sitting atop a pedestal in the first room. Removing the claw awakens a small number of Draugr.
The Ruby Dragon Claw will remain marked as a quest item and cannot be removed from inventory until after Viarmo receives King Olaf's Verse in the quest Tending The Flames.
The Sapphire Dragon Claw from Wilhelm, the Inn keeper of the Vilemyr Inn, in Ivarstead, as a reward for bringing him Wyndelius's Journal, the journal of the man who was "haunting" the Shroud Hearth Barrow. The claw is to be used on the Puzzle door inside the Barrow to gain access to a Word Wall.
Trivia
If a claw is sold to a merchant, it will stay in the shop's inventory indefinitely.
Some of the claws are labeled as "(Material) Dragon Claw," while others are simply called "(Material) Claw." It is unclear if this is intentional. Note: for this article all are labeled "(Material) Dragon Claw."
The Golden Claw has two separate item IDs. The first one (000999E7) is the one recovered from Arvel, and the second (00039647) is if the claw if stolen back from the shop.