Perpetual succession is the term used in accounting in the context of a company. Just as is the case with world which keeps running and it is unaffected by deaths of human beings similarly in corporate world according to this a corporation or company never dies it keeps running forever until it is closed. In simple words even if the owner of a company dies that does not mean that company has died or closed. In case of companies where there is more than 1 owner as is the case with companies which are listed than even if all the owners die then also company will keep functioning.
Perpetual succession is relevant because of business entity concept which states that a company and its owners are distinct and a different person in the eyes of law and therefore a company can keep running even n the absence of its owner.