Short answers, pulled from the story.
In economic theory and econometrics, heterogeneity refers to differences across the units being studied, such as consumers, workers, or firms. A macroeconomic model in which consumers differ from one another is described as having heterogeneous agents.
Unobserved heterogeneity in econometrics occurs when variables that are relevant but cannot be measured are also correlated with the observed variables under study. This correlation causes statistical inferences drawn from the data to be erroneous.
Methods for obtaining valid inferences in the presence of unobserved heterogeneity include the instrumental variables method, multilevel models such as fixed effects and random effects models, and the Heckman correction for selection bias.
The Gorman polar form is a condition on individual preferences that allows individual demands to be aggregated into market demand as if they came from a single representative agent. When preferences satisfy this form and have linear and parallel Engel curves, heterogeneity in preferences can be ignored. If that condition does not hold, a heterogeneous agent model is required.
The distinction depends on what the model assumes about how agents form expectations. Models in which agents have adaptive expectations fall into the category of agent-based computational economics. Models in which agents have rational expectations are dynamic stochastic general equilibrium, or DSGE, models. DSGE models with heterogeneous agents are especially difficult to solve.
Krusell and Smith, publishing in the Journal of Political Economy in 1998, showed that even when the distribution of wealth across agents is arbitrary, prices and equilibrium variables behave approximately as functions of the mean or a few other statistics of that distribution. This insight made heterogeneous agent DSGE models substantially more tractable.