A monopoly is simply a market with only one seller and no close substitutes for that seller's product. Technically, the term "monopoly" is supposed to refer to the market itself, but it's become common for the single seller in the market to also be referred to as a monopoly (rather than as having a monopoly on a market). It's also fairly common for the single seller in a market to be referred to as a monopolist.
Monopolies arise because of barriers to entry that inhibit other companies from entering the market and exerting competitive pressure on the monopolist. These barriers to entry exist in multiple forms, so there are a number of specific reasons that monopolies can exist.
Ownership of a Key ResourceA market can become a monopoly when one firm has sole control of a resource that is necessary for the production of the market's product. For example, the only mud deemed acceptable to rough up baseballs for major league play comes from a particular location along the Delaware river basin, and the knowledge of where this location is is held by a single family-owned firm. This company, therefore, has a monopoly on baseball rubbing mud, since it is the only company that can make a product that is considered acceptable.
Government FranchiseIn some cases, monopolies are explicitly crated by the government when it grants the right to do business in a particular market to a single firm (either private or government-owned). For example, when Amtrak was created in 1971, it was granted a monopoly on operating passenger trains in the United States, and other companies can only offer passenger train service with Amtrak's permission and/or cooperation. Similarly, the United States Postal service is the only company authorized to undertake residential non-rush letter delivery.
Intellectual Property ProtectionEven when the government doesn't explicitly give a single company the right to offer a particular goo or service, it often does so by extending intellectual property protection to companies in the form of patents and copyrights. Simply put, patents and copyrights give the owners of intellectual property the right to be the sole provider of a new product for a specified period of time, so they in essence create temporary monopolies in the markets for new products and services. The rationale behind offering such intellectual property protection is that companies often need such an incentive in order to be willing to undertake the research and development necessary to invent new products and services. (Otherwise, companies might all sit around and wait to copy the innovations of others, and such innovations would never happen. This is, in fact, a specific case of the free-rider problem.)
Natural MonopolySometimes markets become monopolies simply because it is more cost effective to have one firm serving an entire market than it is to have a number of smaller firms competing with one another. Firms whose economies of scale are virtually unlimited are known as natural monopolies, and the goods they produce are referred to as club goods. These firms come to be monopolies because their size and position makes it impossible for new entrants to compete on price. Natural monopolies are usually found in industries with high fixed costs and low marginal costs of operation, such as cable television, telephone, and internet providers.
In all cases, there is a bit of ambiguity surrounding the market definition for determining whether a company is a monopolist. For example, while it is certainly true that Ford has a monopoly on the Ford Focus, it is certainly not the case that Ford has a monopoly on cars overall. The market definition question, which rests on what is considered to be a "close substitute," is a central issue in most monopoly regulation debates.