These threads are also known as the Threads of Life as they represent the life and/or storyline of any one person/creature in the world. Mortals, Gods and Titans all have their own thread, which is why the Sisters of Fate were considered so powerful. They control the fate of the entire world, as they hold the threads of all living beings.
The Threads of Fate are kept with Clotho deep within the Loom Chamber. If one can get a hold of their thread, they can change their own destiny by turning a loom and jumping through the Mirror of the Fates.