The Sin Hearts beat with vile power, and could be used to empower entire demonic hosts, increasing their effectiveness greatly. Visually, they were literally enormous hearts pumping the burning blood.
When Azmodan launched his assault on Bastion's Keep, he dredged the Sin Hearts from the depths of his realm in Hell. Locked away deep within Arreat Crater, the Sin Hearts leant their strength to the hordes battering away at the Keep's gates. Leah got a vision of these hearts, leading Tyrael to identifying them as a threat, and allowing the Nephalem to destroy both.
The Sin Hearts are objects and quest requirements for the quest Heart of Sin. The Heart of the Damned can be found within the Heart of the Damned (area), within the Tower of the Damned. The Heart of the Cursed can be found within the Heart of the Cursed (area), within the Tower of the Cursed.
Both objects need to be destroyed in order to complete the quest. Beware; the Hearts are guarded by a mini-boss. Cydaea, and are shielded by impenetrable barriers until she is defeated. The first time she only teases the player, letting her daughters fight instead; the second Heart, she guards with her life.
Once the protectors have been dealt with, the Hearts are nothing more than mere obstacles. They have no effect on monsters in game, contrary to the lore entry saying that demons are empowered by the mere presence of these artifacts.