HT Games
Quest Rewards
Sometimes, characters in the game will reward you with a Piece of Heart for completing some task for them. These tasks can be as simple as playing a song for somebody -- play along with the Skull Kids in the Lost Woods to receive a heart piece -- or as tricky as one of the harder minigames, in which you get a heart piece from Dampe the Gravedigger by completing the race through his tomb in under a minute.
Chests
Many of the treasure chests in the game contain heart pieces. Some chests are in dungeons, while others are in the open air; in both cases, they'll be well-hidden. Sometimes a chest will only appear after you perform some action -- lighting all the torches in Zora's Domain, for example, will make a chest appear behind the waterfall. At other times, a chest may be somewhere you can't reach: try using your Hookshot to latch onto the chest or some other nearby object.
In the Open
Many Pieces of Heart are neither in a chest nor in the possession of another character -- they're just lying around. One such Piece of Heart, one of the more tantalizing ones, sits atop a pillar halfway up Zora's River. When a heart piece like this one isn't in a chest, you can retrieve it by throwing your Boomerang at it.
Guides
Guides to the locations of every Piece of Heart in the game are extremely common online, and the best ones include illustrative images and detailed descriptions of how to retrieve the heart pieces. If you're having trouble obtaining a particular Piece of Heart, or if you're a "completist" and want to make sure you've found every heart piece in the game, check one of these guides.