When people talk about 'founding' a state, they mean two things:
- Defining a certain territorial region that the 'state' will have sovereign authority to administer and control.
- Creating a body of institutions — offices, laws, systems — by which that region will be administered and controlled.
That can amount to anything, honestly. A few decades back there was a man who took over an abandoned platform in the middle of the North Sea, and declared it an independent country with himself as the presiding authority; he effectively founded a state. The US Founding Fathers did the same thing on a larger scale. They declared that the territory covered by the original thirteen colonies would become a sovereign entity separate from British rule, and then wrote a constitution to define the institutions and systems by which that newly sovereign territory would be administered.
The hard part, of course, is retaining the territory that has been newly declared as sovereign. The people who used to administer it tend to object, violently...
The term 'republic' comes from the Latin res publica (concerns of the people), and is used to signify any state in which the populace as a whole has a significant input into governmental affairs. There is a large variety of systems that qualify as republics, though for the most part they involve deliberative bodies — often of civil representatives — meant to establish governing rules and policies, and administrative bodies meant to implement and administer those rules and policies. The US is considered a republic because it was designed to give the general populace a dominant voice through the election of senators and representatives. Of course, it's worth noting that not every nation that calls itself a republic actually is a republic. The term 'republic' is often used by autocrats who wish to seem more liberal on the world stage: e.g. The Republic of North Korea, or the Syrian Republic. Even the status of the US as a republic is under siege because of gerrymandering, voter suppression, rising authoritarianism, and organized factionalism in political parties, all of which strip the general public of power over their governmental leaders.
And yes, it's all a social construction, but that doesn't make it any less real. We are social creatures, and social relationships dictate the vast majority of our lives. A government merely extends that principle across the limiting boundaries of local community.