The terrain spaces surrounding your City are called wildlands, and the empty ones with nothing different from the bottom are called Plains. When hovering over them with the mouse pointer, it will give information about the benefit it will give you when you have conquered it. Clicking it will give information about current ownership and also give the possibility to scout it out or attack it.
The plain is the only type of wildland that you can build a new city on, to do this you need to first attack it with enough troops to win the battle and conquer it. Then you click the “build city” button to found your new city.
Scouting or attacking a wildland (without a city!) does not break the Beginner's Protection, even if it is currently owned by another player.
You don't get any resource production for having a plain conquered, but you get bonuses for conquering all the other types of wildlands. You can have as many wildlands conquered as the level of your city hall is. Forests are the most popular wildlands to conquer since they give production bonus to your lumber production, which is the most popular resources. Iron is also a popular resource and to get a production bonus here you have to attack and successfully conquer hills. To get production bonus to your stone production you need to conquer the desert type of wildland.