The terrain spaces surrounding your City are called wildlands, and the wooded ones with some coconut trees on are called Lakes. 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 forest of Rock Age is a type of wildland that increases your resource production. The forest increases your resource production of lumber a certain percentage depending on the level of it.
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, Ctystal and Emerald by gathering in Forests.
Like I said before, 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.