Building the Business
In order for a business to form a monopoly, it must first build itself up to such a degree that it can claim a large share of the market. This is incredibly difficult to do, and usually requires vast amounts of capital, hundreds of wise business decisions, several significant acquisitions and mergers and not a little luck. Once a business has found a prominent place on the market "map," so to speak, it is in a position to begin cornering that market.
Buying Up the Market
With its sizable capital, the successful business can now focus on taking over the market completely. One by one, it will buy out its competitors. As it grows, its resources become more and more vast, allowing it to blow away the competition in terms of marketing, not to mention buying power. Soon, no rivals are left to compete with the business--and the business has effectively formed a monopoly. No competition is practically possible.
Another scenario is possible, however. Instead of buying up the market, some companies in some countries begin to curry favor with the governments under which they operate, leading to government-sanctioned monopolies. Often, however, it is the government itself that takes over an entire industry, hoping to reap the benefits of its prosperity. This, too, is a monopoly--and since it is run by the government itself, it is even more impossible for competition to (legally) emerge.