These are coffers containing different treasures that are found in some dungeons throughout Vana'diel. Coffers are typically in higher level areas and have better contents than Treasure Chests. In order to open a coffer, you either need the appropriate key, or need to be a Thief (Main Job) with a lock-picking tool. The keys to open the coffers are dropped from specific mobs, which are usually located in the same zone as the coffer.
When you open a coffer you will either randomly get one of the possible items, some gil, or a quest item if you have a pertaining quest active.
Please note, that while you are weakened, you cannot open a Treasure Coffer.