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The Trapped Chest is a special monster that appears only in the Thief Class Challenges. They have very high attack power, and between their First Strike and Magic Immune traits, there is no way to defeat them without suffering their attack. However, their health is low, so the only challenge is for the player to get enough health survive a single hit from the Trapped Chest, as by that point the player will almost certainly have enough attack power to defeat it in one hit.
Despite the No Experience trait, Trapped Chests are a high priority to kill, as each one drops a glyph or other item, which range from helpful to indispensable in completing the dungeon. The number and levels of Trapped Chests that spawn are not random, and instead remain constant each time the same challenge is run. Their rewards are similarly not random but are predetermined according to the Trapped Chests' levels; see the individual class challenges' pages for full details of which chest contains which prize.
In terms of flavor, the Trapped Chest is not a monster at all, but a treasure chest with loot inside. However, the treasure chest is booby-trapped to grievously injure any Thief who tries to open it and steal the contents. Garrett, the protagonist of the Thief class challenges, seems either to not be skilled enough to disarm the traps or to simply deem disarming not worth his while, and gets through the traps by becoming healthy enough to survive them instead of by avoiding them.