The terrain spaces surrounding your City are called wildlands, and the tender green ones with some grass on are called Grasslands. 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 grassland is the last of the 3 wildland types that will increase your grain production. Many players conquer level 10 NPCs in Rock Age these days, and they then often want to take grasslands around these to get an extra boost in the grain production.
All wildlands increase your resource production apart from plains, which are wildlands you can conquer and build new cities on.
All wildlands can be conquered and gathered for gems except plains.
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 get a better chance to gain Amber, Crystal and Glaze by gathering in Grasslands.
The percentage the wildland will increase your resource production depends on the level of the wildland, but it can also vary between wildlands of the same level too. Number of troops and type of troops that defend each wildland also differs, but generally higher level wildlands have more and stronger troops.