When did the borders among the countries come into being?

Borders came about some ten thousand years ago
when we were tricked from the hunter nomad lifestyle
into the slavery of the agrarian society.

There really weren’t any borders during the middle ages in Europe. But it was a matter of being careful to not trespass on someone’s land without permission.

But no one had documents of any sort before about 150 years ago, unless they were carrying a special letter while on a mission on behalf of a noble.

City-states arose before countries. Germany and Italy did not become unified countries until the 1800s. In earlier times, there were city-states that were often gated, and as @zenvelo points out, feudal properties. So there was the concept of borders in regions, probably even in the ancient world – think of Sparta, Babylonia and the walls of Jerusalem – but not necessarily nations with borders until the late Middle Ages or Renaissance period.

(This is a Euro-centric view, I don’t know as much about Eastern history.)

Often rivers were used as borders e.g. between Texas and Mexico, between Thailand and Laos.

There is a TV series called How The States Got Their Shapes if you want to learn more about the US state borders. You probably know that in the Americas Europeans came over and laid claim to the land and it created borders. Water was often a natural boundary. In the US, over time, pieces of land were bought and sold. The Louisana Purchase was land bought from the French that greatly expanded the US. War changed boundaries back and forth.