Share or stock most commonly refers to a share of ownership in a company. The owners and financial backers of a company may decide to sell the company in order to raise money. However, this usually leads to the loss of control over the company.
Alternatively, by selling shares, they can sell the company, or part of it, to many part-owners. The purchase of one share entitles the owner of that share to literally a share in the ownership of the company, including the right to a fraction of the assets of the company, a fraction of the decision-making power, and potentially a fraction of the profits, which the company may issue as dividends. However, the original owners of the company often still have control of the company, and can use the money paid for the shares to grow the company.
In the common case, where there are thousands of shareholders, it is impractical to have all of them making the daily decisions required in the running of a company. Thus, the shareholders will use their shares as votes in the election of members of the board of directors of the company. However, the choices are usually nominated by insiders or the board of the directors themselves, which over time has led to most of the top executives being on each others boards (see http://www.theyrule.net for an example). Each share constitutes one vote (except in a co-operative society where every member gets one vote regardless of the number of shares they hold). Thus, if one shareholder owns more than half the shares, they can out-vote everyone else, and thus have control of the company.
Shares are usually traded on a stock exchange, where people and organisations may buy and sell shares in a wide range of companies. A given company will usually only trade its shares in one market, and it is said to be quoted on that stock exchange.
A stock option is the right to buy stock in the future at a fixed price. Stock options are often part of the package of executive compensation offered to key executives. Some companies extend stock options to all (or nearly all) of their employees. This was especially true during the dot-com boom of the mid- to late- 1990s, in which the major compensation of many employees was in the increase in value of the stock options they held, rather than their wages or salary. This is still the major method of compensation for CEO's.
The theory behind granting stock options to executives and employees of a corporation is that, since their financial fortunes are tied to the stock price of the company, they will be motivated to increase the value of the stock over time.
The first company that issued shares is considered to be the Dutch East India Company, in 1602.