The Palace of Twilight - Piece of Heart Locations
Like almost every dungeon before it in Twilight Princess, the Palace of Twilight has a total of two pieces of heart for you to obtain. Both of these pieces of heart are found in the first room of the actual dungeon (on two different sides). In order for you to obtain both of these heart pieces you'll either need the Master Sword (to clear some of the fog) or to carry one of the Sols with you, an item that's used throughout much of this dungeon.
Palace of Twilight - First Piece of Heart
This piece of heart is found shortly after acquiring the Sol which is the Palace of Twilight dungeon item. As you back track through the rooms after acquiring the Sol - this chest is found in the very first room, closest to the Twili which are outside.
You will need to bring the Sol with you and use it to lift the darkness in the area. This will allow you to get close enough to the Clawshot platform to Clawshot to it and grab the chest.
Palace of Twilight - Second Piece of Heart
The second piece of heart for you to find in the Palace of Twilight is in the first room (like the last piece of heart) on the opposite side of the dungeon. For this one what you will need to do is carry the Sol with you and take the elevator to the second floor to reach this piece of heart.