To capture an empty square that’s next to a square already occupied by your nation, just click on the square you want to capture (or on mobile, tap it). You can occupy any square of flat land, including beach areas. Nations can’t occupy water or mountainous areas.
In the same way, you can attack a square belonging to another nation, so long as it’s next to a square occupied by your own nation. Just click (or tap) the enemy nation’s square to begin the attack.
Next: The Battle Display