A company -- more specifically its owners and financial backers -- may decide to sell the company in order to raise money. However, if you wish to raise money for use in that company, it is obviously idiotic to sell it to another owner.
If you sell shares, though, you sell the company 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 still have control of the company, and now have the money paid for the shares in the bank to spend on the company.
If you have many thousands of shareholders, it is obviously 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 to the board of the company. 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 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.
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