Two different kinds of monopolies are a pure monopoly and a monopolistic competition. When one company gains control over a specific niche in the market it is generally referred to as a "monopoly."
When one company has control over a specific product and there is no competition, it is called a pure monopoly. In this situation, the company has control of the price, can effectively block competitors from entering the marketplace and can be subject to government intervention. An example of a monopoly is the Bell Telephone Company or the local water or gas company.
When a handful of companies have control over a specific product it is called monopolistic competition. In this setting there are substitutes for a product and prices are dictated by supply and demand.
Examples of monopolistic competition are pizza companies like Pizza Hut and Dominoes or the market between Pepsi and Coke. The differences between the two products have more to do with perception.
In these situations, the customer is a major factor in the price. While there is little difference between the pizzas from either company, and Coke and Pepsi bear a striking resemblance, the way the products are marketed and perceived sets the tone for which becomes the market leader.