For one thing, borders define a nation, giving it an identity and a legal definition. What is Canada or Mexico as nations if there are no borders to recognize them from?
Borders also serve a practical purpose in that they serve to control the flow of goods and people that enter a nation. This allows the country to have security and some measure of control over the economic and criminal activity that happen within those boundaries.
As all societies and nations are made up of laws, and operate under the rule of law, those laws can only apply to a certain geographic region, which of course requires a formal boundary.
In early America with native tribes, borders were less defined and more fluid, and served instead as the practical and operating boundary a tribe could actually defend.
Borders and boundaries are good for countries because without them, the countries could not really exist. The modern concept of a state (most people call them countries, but political scientists tend to call them states) is impossible without the existence of borders.
In modern political science terms, a state can only exist if it has a set boundary. Within that boundary, it must be sovereign. If there are no boundaries or borders, states cannot be sovereign because they will not know where their laws are in effect and where others' laws are.
When borders exist, countries can exercise sovereignty. They can (in theory) control who may enter their country. They can enforce their laws within those borders. Because of this, borders are central to the entire concept of the state.