a corporation as a person

     There has been some recent noise about a corporation being a person. I missed what started the fray--but, in some respects, a corporation is a person. For the purposes of lawsuits or financial responsibility, a corporation is considered a person. No one can sue the officers of a corporation for an accident, for instance. The party named in the lawsuit is the corporation, and if at fault, it is the corporation that is expected to pay.  In legal terms, a corporation is a "legal person".  This is why businesses incorporate. People put their money into a corporation--even if it is a small family business. Their homes and personal property are not at risk of being claimed for a debt the corporation owes.

a share of stock

     A share of stock is actually so easy to understand as to make the level of confusion surrounding it suspicious--as in perhaps it IS all a diabolical plot. That said, a share of stock is a part interest in the corporation that issued the stock. If a publicly owned (stock-issuing) corporation issues on million shares of stock, and you buy one of them, you have purchased a millionth part of the corporation. You are a part owner--one part in every million of that corporation is yours.
     The stock in publicly owned corporations (remember that means stock issuing, not government owned) usually comes with a vote--one vote per share of stock. It's not exactly a democracy, which would have one vote per person--it's just how a corporation is run. Stockholders elect the board of directors of the corporation. The board of directors is headed by a chairperson of the board, or a chief executive officer (CEO).
     If you own stock through mutual funds, you have probably given your vote as a "proxy vote" to an employee of the mutual fund ( a mutual fund buys and sells shares of stock, and then sells shares in itself to investors--much like "holding company" of the 1920's, now illegal).