There is 36 trophies, 16 of them are related to the story of the game (thus their descriptions in the PlayStation system are initially hidden). They are automatically unlocked during the campaign. With the completion of the game, you will gain all of them.
Releasing the Floodgates
Description: you will get this trophy at the end of the prolog section, after killing the first boss - Poseidon. You cannot complete the game without getting it.
Burnt to a Crisp
Description: during the level in Hades you will come across a man imprisoned behind dead thickets. Set fire to both him and those plants and possess Apollo's Bow. That will unlock the described prize.
Description: during the level in City of Olimpia you will come across the god of Sun - Helios. Kill him and get the trophy.
Description: near the half of the game you will reach Olimpian Citadel and you will meet Hermes. Kill him - obtaining his boots - and unlock this reward.
I didn't do it...But I wish I did!
Description: after reaching Poseidon's Chamber you will come across a woman of the god of seas. Take her through the corridors and use as a holder of the revolving mechanism which rises the bars. This will make that she will die, further way will be available and you will get the trophy.
Description: this is the only story trophy which can be obtained during the campaign. After reaching Aphrodite's Chamber for the first time, a cut-scene will be triggered. During the scene the goddess will invite you to her bed. After that a contextual menu will appear - choose 'yes' and play with her (QTE). The success will give you the trophy.
In order to do it, first, press three times basic buttons. After that turn the left stick a few times and as a result shake it. At the end press the circle button 6 times and finish by mashing it
Description: in order to take the possession of this trophy you have to reach the entrance to the City of Olimpia. Near the gate the well-known-titan will cross your way. Cut out her palm and push her into the precipice.
Description: in order to get this trophy you just have to finish the level - Gates of Thisiphone. Thanks to that you will reach the Tartarus.
Description: an objective which completed will unlock the trophy is rescuing Pandora form the Daedalus' Labyrinth. This level is rather long and the reward will be unlocked at the very end of it, on the top of the cubes.
Three Wise Men
Description: the trophy will appear as unlocked after breaking the Chain of Balance, during the level Judges of the Underworld part 2, near the end of the game.
Description: defeat your brother in life-and-death battle during the level Forum - about the half of the game.
Description: in order to unlock the trophy you have to kill the Cronos. You will come across him in the Pit of Tartarus.
Description: the trophy will be unlocked after killing Zeus, at the end of the game. The battle will be multiphase and the reward will be gained at the end.
Description: in order to get this trophy you have to kill Hades. The battle will be multiphase and only when you finish the last one you will achieve a desired effect.
Description: in order to unlock this reward you must kill a big scorpion during the second visit in the Caverns (near the end of the game) and grab the item left on the arena of the battle.
Seeing things from a different perspective
Description: near the end of the game you will reach Hera's Garden. Accomplish it and get the reward.