1. Natural monopoly - Returns to scale are large enough (or the market it small enough) that one firm can provide the product for a lower price than multiple firms. This is possible without government backing, but is basically harmless if entry into the industry is not blocked. The natural monopoly may be the only seller, but it still faces potential competition.
2. Artificial monopoly - Firms merge to monopoly, firms form a cartel, etc., in an industry without returns to scale. This is the kind of monopoly that people worry about, but in a free market, they are (or would be) very uncommon. There is always an incentive to cheat on a cartel agreement, and there is always an incentive to enter the industry if profits are above normal.
3. Government monopoly - Firms take advantage of government-created (whether intentional or unintentional) entry barriers to establish an artificial monopoly, or the government itself establishes a monopoly in the activity. Very common. For instance, heavy regulation might favor large firms that can more easily absorb the fixed cost of compliance.
If you are interested in learning more about this, look into the branch of economics called "Industrial Organization." The best textbook is by Carlton and Perloff; it is the one I used when I taught IO this semester.