The terrain spaces surrounding your City are called wildlands, and the desolate ones with some cactuses on are called Deserts. 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 desert is actually where you will get your stone production bonus from, not from the hill as many players think. The desert is not that popular to conquer however, as stone is pretty much the least popular resource in Rock Age these days.
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 Glaze, Jade and Agate by gathering in Deserts.
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.