Chain of Fate

Souls
The Chain of Fate is what binds souls to their living body. When the Chain of Fate is broken, an erosion will take place from the end of it after being broken. Once this happens a gradual process of hollow transformation takes place. A sign of being a step closer to becoming a Hollow is when the soul with a broken Chain of Fate starts to feel hungry. The erosion of the last piece of the Chain of Fate is far more painful than that of the previous ones(the Last Erosion.) Ichigo was able to fight the Hollow transformation but gained Hollow powers inadvertently anyway. Interestingly enough, Ichigo Kurosaki's bankai Tensa Zangetsu has a chain attached to the hilt resembling the Chain of Fate.

For more information on the Chain of Fate's erosion, see Encroachment

Shinigami
The Chain of Fate on a Shinigami is just as important as it is on an plus. While there is no chain per se connected to a Shinigami, the area where the Chain of Fate was originally is still just as important as when the Shinigami was a simple soul. In a Shinigami the Chain of Fate acts as a boost to spiritual power in conjunction with the Soul Sleep which is the source of spiritual power. These two points are commonly considered pressure points. If they are pierced they seal up this spiritual power, and the Shinigami will loose all their power. The Shingami who experiences this will never be a Shinigami again.