Short answers, pulled from the story.
Economic equilibrium is a condition in which the forces of supply and demand are balanced, so that economic variables stop changing. At the equilibrium price, the amount of goods buyers seek equals the amount sellers produce; this price is often called the competitive price or market clearing price.
Huw Dixon proposed three properties: P1 (consistency of agent behavior), P2 (no agent has an incentive to change its behavior), and P3 (the equilibrium is the outcome of some dynamic process, meaning it is stable and reachable). All three are satisfied in a competitive equilibrium but P1 and P2 fail in a monopoly equilibrium.
The first use of the Nash equilibrium in economics was in the Cournot duopoly model, developed by Antoine Augustin Cournot in his 1838 book. Cournot modeled two firms producing an identical product and choosing outputs to maximize profits given the other firm's output.
Food was exported from Ireland during the Great Famine of 1845-52 because the equilibrium price of the Irish-British market for potatoes was above what Irish farmers could afford to pay, making sales to English buyers more profitable. Paul Samuelson cited this as a reason economists should not attach a moral or value judgment to the equilibrium price.
In a static equilibrium, all quantities have unchanging values. In a dynamic equilibrium, quantities such as output and the capital stock may all be growing, but at the same rate, leaving their ratios unchanging. The neoclassical growth model is a standard example of dynamic equilibrium.
Comparative statics is the process of comparing two static equilibria to each other, examining how a change in one variable shifts the equilibrium price and quantity. For example, a rise in consumers' income leads to a higher equilibrium price, so the comparative static effect of consumer income on price is positive.