Open Borders is an agreement with another leader to allow them to move their units, military or civilian, through your territory. Without this agreement, entering another leader's territory is an act of war, except with Religious Units, who can ignore borders entirely. If you have Open Borders with a leader, their border will appear as a dotted line instead of a solid one. Open Borders is unlocked with the Early Empire Civic. Prior to that, your borders are open to everyone.
Open Borders can be offered as part of a Deal trade agreement if both parties have researched the Early Empire Civic, but it is offered individually by each leader—you can be granted Open Borders with a leader without granting them Open Borders in return. The deal lasts for 30 turns, and then it needs to be renewed.
If you are in an Alliance with another leader, you both grant each other Open Borders automatically.