Connections are bonds between people or objects and can be emotional or physical or anything really as long as there is something connecting them, a relationship. By manipulating them, you can tie the circumstances from one thing to another in a variety of ways. By cutting them you can damage or even erase the relationship between the connecting points.
Connections also tie people and things to the world, so if all the connections get severed then they fall into the cracks of Limbo. When seen through by a practitioner they appear as ribbons formed out of spirits.