"The State" is a theoretical concept in social science. It has two common definitions, "a body of armed men," and "the method of one class's political repression of all other classes." The definition being used in your world history text is the first definition.
A strong state is one which then fulfills the core requirements of maintaining a body of armed men in its circumstances. Stronger states will maintain a body of armed men in a way which makes the dissolution of that capacity less likely. This isn't just numbers or quality, but also the soft power that results in population growth, legitimacy and capacity to mobilise, the ability to bind outlying semi-governed polities beneath a central or networked will. Less strong states will be susceptible to invasion, successful or unsuccessful revolution or revolt, mass resistance to basic policies (tax, land, family structure) or esoteric policies (customing of luxury goods, public nuisance, courtesy within a single class).
Strong states possess the possibility to impose their will internally and externally, in the final instance by force. Weak states lack this capacity.
Not all "weak states" are necessarily weak countries or nations. A state is not a country, nor a nation, nor a revolutionary class. French states during the great revolution were weak until Napoleon, with rapidly changing policy aims, and a comparatively disorganised military. New innovations in nationalism and non-state networks of power allowed the communities and community of France to enact decisions, despite the weak state. Similarly with these early United States.
Using the second definition, not present in your textbook, a strong state is one that would ensure continued class rule by the socially dominant class. While this obviously has elements of defence of the state against external threats, states are quite willing to fold as long as the system of class rule is maintained (Franco-Prussian War, the English crown's possessions in France). Correspondingly, this second definition of a "strong state" places more emphasis on capacity for internal class repression, or repressive desublimation. This can mean cross-class alliances. A classic example would be the post-war welfare state / full-employment / low inflation nexus in capitalism. This reduced profit and increased the social strength of the working class, but also reduced working class militance. This kind of state extends outside of formal government, and into hegemonic apparatus and the workplace itself.
