As defined by Wikipedia, "A stock exchange is a mutual organization which provides "trading" facilities for stockbrokers and traders, to trade stocks and other securities. Stock exchanges also provide facilities for the issue and redemption of securities as well as other financial instruments and capital events including the payment of income and dividends. The securities traded on a stock exchange include: shares issued by companies, unit trusts, derivatives, pooled investment products and bonds. To be able to trade a security on a certain stock exchange, it has to be listed there. Usually there is a central location at least for recordkeeping, but trade is less and less linked to such a physical place, as modern markets are electronic networks, which gives them advantages of speed and cost of transactions. Trade on an exchange is for members only. The initial offering of stocks and bonds to investors is by definition done in the primary market and subsequent trading is done in the secondary market. A stock exchange is often the most important component of a stock market. Supply and demand in stock markets is driven by various factors which, as in all free markets, affect the price of stocks".
An off exchange or over the counter exchange is merely stock that is not issued or traded on the stock exchange. Bonds and derivatives are usually traded in this manner.
A company may use a stock exchange as a means to raise capital through selling shares to the investing public. Alternatively, a company might see a potential acquisition as an opportunity to expand their product lines or increase it's market shares and a takeover bid or a merger done through the stock market is one of the most common ways of facilitating this type of transaction.
People that invest in shares instead of putting their money into saving accounts are seen to be promoting business activity such as energy utilities, agriculture, commerce and industry by allocating resources that would've normally just be sitting in low interest accounts.
Stock exchanges allow for both the new and experienced investor to share in profitable businesses through dividends and stock price increases which result in capital gains. Likewise, if an investor does not have a huge capital outlay, investing in shares on the stock exchange can allow one to buy only the number of shares they can afford. Therein, stock exchanges can facilitate the means for smaller investors to own shares of the same companies as bigger investors.