There are ten Godly Possessions to be found in the game. You don't have to get them all in one playthrough. Hera's Chalice, the final Godly Possession, must be picked up during the storyline. Below is a YouTube video (credit to Rooster Teeth's Achievement Hunter), if you'd prefer that instead.
1) Zeus' Eagle
Found in the Heart of Gaia during the beginning of the game. While inside Gaia, you'll eventually come across her heart and you must move a part of the wall from the right to the left; before doing this, climb the vines to the right of the room to the top and pick the item up.
2) Hades' Helm
After defeating Hades, the second boss of the game, this can be found at the bottom of the circular pool in which he stood during the final phase of the battle; press square to dive and swim towards the camera while underwater to find it.
3) Helios' Shield
After ripping off Helios' head (not long after using the ballista to destroy his chariot) in the City of Olympia, his shield can be found to the right of this small area.
4) Hermes' Coin
Once Hermes is dead, this can be found lying behind the head of the large statue of Athena at the back of the room; it is hidden by the head, but the camera will move and reveal it once Kratos goes behind it.
5) Hercules' Shoulder Guard
Similar to Hades' Helm, this is found underwater after defeating the respective god: after defeating Hercules, dive to the bottom of the pool that Hercules' body and Kratos fall into to find it.
6) Poseidon's Conch Shell
This is found in the same room that Poseidon's slave/lover is in (you can't really miss her) which can be accessed through a portal after solving the dog puzzle. It's in an alcove at the back-right of the room.
7) Aphrodite's Garter
Found in the back of Aphrodite's Chamber (Aphrodite and her two lovers are making love on the bed - again, you can't really miss them). This is an easy Possession to miss, you need to jump from the right of the bed and glide around behind it. There is a bottomless pit nearby, so be careful!
8) Hephaestus' Ring
Found after killing Hephaestus, not long after killing Cronus. Once Hephaestus is dead and you're properly introduced to your new weapons, head to the left of his body to find the massive ring next to the debris that covers the entrance to Tartarus.
9) Daedalus' Schematics
This is found in Daedalus' Workshop, just past Aphrodite's Chamber. Before exiting the room, fire the ballista through the portal behind the gate you just opened and follow it through. Pulling the lever here will temporarily lower two chests; you must quickly head back through the portal and fire the ballista again before the chests are raised.
When done successfully, the chest on the left has the schematics.
10) Hera's Chalice
This must be picked up as part of the story, just at the entrance to Hera's Garden.