Methodological individualism (MI) is a doctrine in the philosophy of the social sciences about the relationship between "society" and individuals. The idea can be formulated in several related but somewhat different ways: social facts are constituted by facts about individuals; social entities are composed of individuals and their properties and relations; social structures and entities are "nothing but" ensembles of individuals and their behaviors; social explanations must be derivable from facts about individuals; scientific statements about "society" must be reducible to statements about individuals and their properties and relations; social laws or generalizations must be derivable from general facts about individuals. And there are probably other possible formulations as well.
So the doctrine of methodological individualism often represents a form of reductionism from one area of scientific theory to another: theories about social entities and properties must be reducible to theories and statements about individuals and their properties. This line of thought is parallel to materialism in the philosophy of mind or anti-vitalism in the philosophy of biology. Mental states must be reducible to a set of facts about neurophysiology; statements about living organisms must be reducible to a set of facts about the molecular chemistry and physiology of cells. (See Ingo Brigandt and Alan Love's article in the Stanford Encyclopedia of Philosophy on reduction in biology.)
A somewhat less restrictive view than reductionism is the theory of "supervenience" between levels of scientific description. According to the theory of supervenience, facts at one level of description are fixed or determined by facts at a lower level of description. To say that X supervenes upon Y is to say that there is no difference between states of affairs concerning X for which there is not also a difference in states of affairs concerning Y. This is a less restrictive doctrine because it doesn't require that we provide derivations of the facts of X from facts of Y. (Jaegwon Kim is the primary innovator here; see Physicalism, or Something Near Enough for a recent formulation of the theory.)
Methodological individualism is the limit version of a family of perspectives on social explanation that we might refer to as "agent-centered" approaches to social explanation. Here the general idea is that we explain social outcomes as an aggregate result of the actions, choices, and mentalities of individuals. Individuals' behavior and choice constitute the causal dynamics of social outcomes. The idea of "microfoundations" has played an important role in recent thinking about the relationship between social facts and individual facts. (See Microfoundations, Methods, and Causation for more on the idea of microfoundations for social explanations.)
But agent-centered approaches can give more "social-ness" to the individual than the founding statements of MI would permit. For example, the position of "methodological localism" identifies socially constituted and socially situated individuals as the foundation of social explanation; but this position explicitly denies the idea that all social facts are reducible to bare psychological facts about individuals. Rather, individuals are themselves constituted and constrained by previously established social conditions. (See "Levels of the Social" for more about methodological localism.)
The idea of methodological individualism is one that has appealed to philosophers and social thinkers for almost as long as there has been systematic thinking about social science. Modern philosophy of social science began in the nineteenth century, and John Stuart Mill's theories of social knowledge contained the assumption of methodological individualism (The Logic of the Moral Sciences). Max Weber also put forward the doctrine in The Methodology Of The Social Sciences. A classic statement was presented by J. W. N. Watkins (1968), "Methodological Individualism and Social Tendencies" in May Brodbeck, Readings in the Philosophy of the Social Sciences.
The logical contrary of MI is the idea of social holism, most explicitly advocated by Emile Durkheim in Rules of Sociological Method. Holism is a form of anti-reductionism; it maintains that there are facts about the social world that do not reduce to facts about individuals. Society is autonomous with respect to the individuals who "make it up". There are social forces (e.g. systems of norms) that exercise causal power over individuals, instead of norms being constituted by the psychological states of individuals. Other varieties of social holism are possible as well. Structuralism is the view that social structures exercise autonomous causal properties, as expressed in such authors as Levi Strauss, Althusser, and Foucault.
Arguments in favor of methodological individualism derive from several insights. First, there is the point that social facts are evidently constituted by the thoughts and behaviors of groups of individuals. Social movements are composed of individuals with specific psychologies and beliefs; organizations are composed of individuals; and, arguably, moralities and cultures are made up of individuals with specific beliefs and values. Second, there is the point that social "laws" are rare, exception-laden, and conditional; so there is a methodological reason to look for the more basic laws that may regulate social behavior -- at the level of individuals and their psychology. Third, there is a preference for ontological sparsity: if we can explain social facts in terms of facts about individuals, then we don't need to attribute ontological status to social facts and entities. Fourth, there is a "materialist" or anti-occultist preference that is appealing to many philosophers; the idea that social facts might be autonomous with respect to individuals gives an impression of occult causal powers or action at a distance. So there is a range of reasons to think that social outcomes are made up of or determined by the aggregate results of individuals and their interactions.
MI has been particularly appealing in certain disciplines of the social sciences -- especially economics and political science. In each case the perspective of "rational actor theory" has appeared to be a very promising line of explanation: explain the behavior of groups (consumers, voters) as the aggregate outcome of individuals making choices with a specific set of beliefs and preferences, within a specific set of constraints.
Other areas of the social sciences are less amenable to the theory of methodological individualism. Anthropology and sociology are disciplines that set the focus on the higher-level social conditions or causes that influence behavior -- a perspective that seems to be more comfortable with some form of holism. However, there is a range of opinion among practitioners of these disciplines as well, and there are anthropologists and sociologists who are more sympathetic to the impulses of methodological individualism.
It is important to highlight some points that MI does not entail. MI does not entail that individuals are egoists or purely self-regarding. It does not entail that individuals are not social. It does not entail that social facts do not have causal consequences -- for other social facts and for individual behavior. It is indeed possible to reframe almost all substantive sociological theory in terms that are consistent with the reasonable conditions implied by MI. Even Durkheim's central theories can be formulated in a way that is innocent with respect to the charge of "action at a distance". And, from the other direction, even a theorist with as clear a commitment to MI as Max Weber, is still able to make "macro" or "holistic" claims about the causal importance of factors such as religion or morality.