It occurs in a market where the average cost curve is decreasing when it intersects with the demand curve. In standard supply and demand theory with a perfect competition this creates an equilibrium price that is below the average cost to produce.
In particular, companies that grow to take advantage of economies of scale often run into problems of bureaucracy; these factors interact to produce an "ideal" size for a company, at which the company's average cost of production is minimised. If that ideal size is large enough to supply the whole market, then that market is a natural monopoly.
Some also use the term for suppliers in a market for which entry into the market is extremely expensive. For example, early railroad and telephone companies rarely had to worry about competitors because they would have to duplicate large amounts of track or wiring that the earlier company had already paid for. In this case, the monopolies can be eliminated either by new technologies like the automobile and wireless communications, or by government intervention.
Public utilities are frequently viewed as natural monopolies. This is considered a justification for government recognition and regulation of a single supplier in each such market. If it is a natural monopoly, then (it is argued) allowing or expecting the action of competition is bootless.
In many cases, established suppliers have lobbied governments to be treated as natural monopolies -- despite the burden of regulation. They do this to ensure a stable price, enable a constant return for their shareholders, and reduce the risk of competition. Opposing this trend is utility deregulation, in which suppliers (usually at the request of government regulators) seek to reduce the burdens of regulation while accepting some competition in what they may have formerly held to be a natural-monopoly market. Manipulation of the image of natural monopoly may in this way be used in rent seeking as suppliers attempt to obtain a regulatory environment to their advantage.
Software is often taken to be a natural monopoly, due to the high cost of making the first copy and the low cost of replication. These factors create an average cost curve that typically decreases for any quantity greater than one. This argument has been used to justify arguments relating both to Microsoft's current personal computer software market domination, and to suggest the possibility of its replacement by a future natural monopoly of free software.
Note that network effects are considered separately from natural monopoly status. Natural monopoly affects are a property of the producer's cost curves, but network effects are a property of the benefit to the consumers of a good. Many goods have both properties like operating system software and telephone networks.