Short answers, pulled from the story.
In economics, an agent is a decision maker in a model of some aspect of the economy. Agents are typically assumed to solve an optimization or choice problem, and they may represent consumers, producers, households, firms, governments, or central banks.
A representative agent model assumes all agents of a given type are exactly identical, allowing for the simplest possible description of the economy. A heterogeneous agent model recognizes differences among agents, such as differences in age or wealth, and is used when those differences are directly relevant to the question being studied.
Macroeconomic models, especially dynamic stochastic general equilibrium models built on microfoundations, commonly distinguish households, firms, and governments or central banks as the main agent types. Some models add further distinctions, separating workers from shoppers or including commercial banks as a distinct agent type.
In principal-agent models, the agent refers specifically to someone delegated to act on behalf of a principal. The principal is the party on whose behalf the agent acts, and the models examine whether the agent's interests align with those of the principal.
In agent-based computational economics, agents are computational objects modeled as interacting according to rules over space and time, not real people. The rules are formulated to model behavior and social interactions based on stipulated incentives and information, and the concept may be broadly interpreted to include any persistent individual, social, biological, or physical entity.
Economists use heterogeneous agent models when differences among agents are directly relevant to the question at hand. For example, a model studying the economic effects of pensions likely needs to account for differences in age, and a model studying precautionary saving or redistributive taxation likely needs to account for differences in wealth.