The Damocles Key is a key item in the Tomb Raider level St. Francis Folly. Lara can find it in a chamber dedicated to Damocles, a figure in Greek mythology.
Inside the chambers Lara is confronted with big swords dangling from the ceiling overhead. At first they appear to be harmless but once the key has been picked up, they start coming down when Lara is standing underneath.
Just like the other three keys (the Atlas Key, the Neptune Key and the Thor Key) it will open one of the four metal bolts that keep the gate to the Colosseum shut.