They may be bought and sold in the stock market if the corporation is "publicly traded" (i.e. stocks can be purchased by any person), or may be "privately traded" (i.e. the company is held by only a few people, often family members of founders or later purchasers, and cannot be sold to anyone without the approval of the other owners; an example is the large Florida grocery chain Publix, which limits ownership to its employees).
Stock may be either "common" or "preferred". The difference is that common stock has voting rights attached (though a company can have multiple classes of common stock, with some having more voting rights than others: an example is Berkshire Hathaway has both Class A and Class B stock where the Class A stock has 10,000 times the voting power of the Class B stock), while preferred stock has preferential rights in receiving dividends over common stock (Public Storage, a large owner-operator of self-storage facilities, has multiple classes of preferred stock, each paying a different dividend amount).