National Borders are the bold lines at the outline of your particular color on the map. They symbolize the end of your territory; you cannot build outside of your territory (with the exception of the Lakota if the area is not controlled by any opposing nation).
To expand your territory, you need to build cities or Forts close to your borders. Once the city is completed, your borders increase in that particular area. If the entire map is occupied by other nations, you have to either ally with a specific nation to construct additional buildings outside your territory or resort to war.