The first market structure is perfect competition. Economists argue (as in the link below) that there is no industry that truly meets the definition of perfect competition. However, they say that agricultural products are as close as we can come to such a market. For example, Grade A eggs from one farm cannot really be differentiated from another farm and it is relatively easy to become an egg farmer.
Monopolies are often granted by the government. An example of this would be the company that provides electricity in your town. It's the only provider and no other firm is allowed in.
Monopolistic competition is seen mainly in retail and in restaurants. If you live in a town of any size, there are probably at least a few Chinese restaurants. These are in monopolistic competition with one another and, you can argue, with other kinds of restaurants as well.
An example of oligopoly is the auto industry. There are only a few big players in the auto industry. They sell products that they try to distinguish from one another and it would not be at all easy for a new company to enter the market.