The Chain of Fate plays a part in the form a spirit will take on after death. A spirit whose regrets revolve around a person has its chain bound to that person and becomes an "obsessed spirit"; if the spirit's regrets involve a location, or if some incident happened at that location which caused its death, the spirit's chain will be bound to that location and it would become an "earthbound spirit" or Jibakurei.
A person who has had their spirit removed from their living body will experience a great pain coming from the chain and might feel the need to break it. Breaking the chain, however, would render the soul unable to return to its physical body. A Shinigami can return a spirit to a body using healing Kidō if the chain is intact and if the separation hasn't lasted for too long.
Encroachment or corrosion is the process of a Plus gradually losing its Chain of Fate. The process physically begins the moment the body dies and the Chain of Fate is broken. The Encroachment usually lasts months or even years, although a special gas exists that can speed up the process to a mere 72 hours. With every cycle of the Encroachment, the furthest link in the chain begins to gnash and chew into itself and the links attached to it. The process cannot be halted by any known means - any attempt to do so would end in a nasty bite.
During each cycle, the Plus experiences excruciating pains, the worst of which come with the final cycle which is much more powerful than those before. During this cycle, all the remaining links undergo the Encroachment at once.
For Ichigo, it was possible to resist this process during the Trials of the Shattered Shaft due to his latent Shinigami powers. It was Zangetsu who pulled Ichigo into his internal world and taught him how to tap in to those powers. However, since Zangetsu's intervention came after the Encroachment was physically through, an inner Hollow, similar to those of the Visored, was formed in Ichigo's internal world, manifested by a Hollow mask created over Ichigo's face.
The Saketsu (鎖結, Binding Chain) on a Shinigami is just as important as the Chain of Fate is on a 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 Saketsu acts as a boost to spiritual power, in conjunction with the Hakusui (魄睡, Soul Sleep), 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 lose all their power. The Shinigami who experiences this will never be a Shinigami again.