Short answers, pulled from the story.
Imperfect competition is a market that fails to satisfy every condition required of a perfectly competitive market, which causes market inefficiencies and can lead to market failure. It is measured by how far a firm can push price above marginal cost.
The four broad market structures of imperfect competition are monopolistic competition, oligopoly, duopoly, and monopoly. They differ by the number of sellers, the degree of product differentiation, and the degree of control firms have over price.
A market is considered imperfect if firms are not price takers and control their own pricing, if there is only one seller or none, if there are barriers to entry and exit, if there is information asymmetry between buyers and sellers, or if goods and services are heterogeneous or differentiated.
The Herfindahl Index sums the squared market share of every firm in a market, producing a value between 1 divided by the number of firms and 1, where a higher value signals a more concentrated market. Perfect competition and monopolistic competition sit below 0.20, oligopoly sits between 0.20 and 0.60, and monopoly sits above 0.60.
Market power in imperfect competition comes from control over important inputs, copyrights and patents, network economies such as Instagram's growing popularity, and government licenses such as Yosemite Hospitality's license to run a lodge in Yosemite National Park.
Economists disagree because theorists who favor the imperfect view argue that policy based on perfect-competition assumptions is ineffective, since no market is ever purely perfectly competitive. Others still favor perfect-competition assumptions because their logic is widely used and consistent imperfectly competitive models remain lacking.