Upon founding a new city, all tiles around the City Center are claimed, plus additional tiles extending outward. For Russia, their unique ability greatly amplifies this effect. Note that it can't claim tiles of another civilization.
Initially, borders will be open and weakly defined, enabling units from other civilizations and city-states to enter. Once the Early Empire civic is researched, borders will switch to closed. Unless an Open Borders agreement is made, attempting to enter closed borders constitutes an act of war. For city-states, Open Borders is granted to players that have reached suzerainty status or slotted the Gunboat Diplomacy policy card.
Note that Traders and religious units are exempt from these rules, and may pass into closed borders. This means Missionaries and Apostles can be used to perform reconnaissance into foreign lands. Additionally, the Terracotta Army wonder grants Archaeologists the ability to enter closed borders.