A Perfectly Competitive Market is characterized by a large number of buyers and sellers, here the prices are determined by the demand and supply forces. It is important to note that the sellers sell only the homogenous and related goods under this type of competition.
Every buyer has a perfect knowledge about the prices, costs, etc.. And, there is a free flow of factors along with freedom of entry and exit of the firms in the market.
Monopolistic Competition markets are characterised by a large number of sellers just like perfectly competitive markets. However, one major difference is that the goods sold in these type of markets are not homogenous. The products may be substitutes but not perfectly.
The prices are set by the sellers themselves, as there may not always exist competition between the sellers (because of non-homogenity of goods). There is free entry and exit of the firm because a firm thrives because of it's own capabilities and competitiveness in these type of markets.
Pure Monopoly refers to a situation where there may be many buyers but only one seller. The monopolist (the single seller) is the price maker and determines the price as per his convenience.
The type of product or services that a Monopolist sells or renders may be different from the rest of the market. There may not exist perfect substitutes to it's products or services. Free entry and exit of the firms is not possible because with that Monopoly would cease to exist.
In a Duopoly there exist only 2 sellers. The two firms mostly sell the same type of products and there exists high competition among them.
The action taken by one firm is crucial for it's competitor and so does the determination of the prices (it depends on their competitor). Free entry or exit of the firms does not exist because then it would no longer be a duopoly market. If any other firm(s), apart from the two competing firms, joins the competition, it leads to Oligopoly and if any one firm leaves it leads to Monopoly.
An oligopoly market is characterised by few sellers (definitely more than 2) that are generally huge in size. Just like Duopoly the price determination or any other policy formulation depends on competitors because of the existence of competition.
The free entry of the competitor firms in such markets is difficult because of many reasons like;
i. it would be difficult for a new firms to compete with the giant firms of oligopoly.
ii. oligopoly firms may have their own segment of loyal customers.
iii. oligopoly firms are huge and have the ability to set the prices in such a way that it would make the entry of new players impossible.
It is also worth noting that if oligopoly firms work in unison and harmony it would lead to the existence of Monopoly type of market characteristics.