The terrain spaces surrounding your City are called wildlands, and the green ones with some grass and water on are called Hills. 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.
Swamps are another type of wildland that will increase your production of grain by a certain amount. They are not that popular for players to conquer however, as they usually give worse percentage increase of resource production compared to the lakes you can also conquer.
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 gatherred 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 Emerald, Pearl and Coral by gathering in Swamps.
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.