The best one-sentence summary of Huntington's book I can think of is an update of Aristotle's Politics for the 20th century. Huntington's "big point"--the most important source of political variation between countries is their degree of government, not their type of government--is straight out of Aristotle's distinction between proper and degenerate regime types. A lot of people read Huntington as a conservative and a defender of elite privilege, but in this book at least Huntington is not really making a liberal/conservative point. It's a political point to be sure, but the political agenda commonly associated with Huntington is narrower than the one he's really pursuing here which is an institutionalist, elitist, gradualist conservatism against most forms of radicalism.
Huntington's arguing against a once widely-held view in political science called "modernization" theory that held, to simplify greatly, that countries were on a smooth escalator of development in which democratization, economic development, and the emergence of a modern society all move together. Huntington argues against modernization view that modernization misses a vital necessary condition of political institutionalization. Without political institutionalization, modernization actually leads to political decay and instability rather than political order. Political institutionalization refers to the growth in complexity, adaptability, autonomy, and coherence of political organizations like parties, the military, the bureaucracy, the executive, the judiciary, and the legislature. Political institutionalization, Huntington claims, allows political organizations to mediate the social and economic forces unleashed by development and modernization and pursue the common interest rather than the interests of particular factions (again echoing Aristotle).
Huntington's basic evidence for his critique of modernization is increasing political instability in developing countries during the 1950s and 60s in the form of coups, revolutions, and protests. Rather than a smooth monotonic relationship between modernization and political stability, Huntington argues instead for a kind of political Kuznets curve in which political stability is initially decreasing in modernization and later increasing.
It would be a disservice to this book not to acknowledge how conceptually rich and provocative it is. Huntington's conceptualization of institutionalization is fascinatingly rich as well as his discussion of political decay in what he calls "praetorian" (non-institutionalized) polities. The big problem with Huntington's argument, however, is the claim that the quality of political institutions primarily explains the a country's level political stability. Basically, while Huntington's critique of modernization theory is right, his proposed missing variable is not the one. It's worth noting that Huntington only argues that highly institutionalized organizations are /capable/ of pursuing the public good, not that they actually do. As he says, "The capacity to create political institutions is the capacity to create public interests" (24). Note that this says nothing about how these institutions are /actually used/. More fundamentally, however, Huntington doesn't give an argument for why the quality of political institutions rather than the underlying /determinants/ of quality itself. For example, Tony Marx argues, one confounding variable for Huntington is racial exclusion. Racial exclusion, which leads to racial homogeneity among the included citizens in a polity, can be a pre-requisite for political institutionalization but also a direct determinant of political stability.
A different objection to Huntington is the possibility of reverse causation. It may be that political stability itself determines the level of political institutionalization in a country and not the other way around. Huntington doesn't give a real argument for treating political institutionalization as the explanatory rather than the dependent variable.
All this is to say that Huntington draws a possibly useful conceptual distinction between two classes of regimes and makes an association between the two, but he doesn't actually show that there's a causal relationship between the two. Huntington's "stylized fact" that politically stable countries tend to be either very rich or very poor, very "modern" or very "backward," may be right, just as Lipsett's famous stylized fact that all rich countries are democracies may be right. However, we want more than associations--we want explanations, and causal explanations are good explanations.