Okay, this might sound like a stupid question but, is there some sort of a permission in the US I might require to fly a quadcopter or a UAV for that matter? I couldn't find much help anywhere else.
For the US law, small UAVs are analogous to RC models and the regulation depends on the use intent.
For non-commercial, recreational use, the law is relatively permissive and mostly relies on the rules of the Academy of Model Aeronautics. You can find their rules here: http://www.modelaircraft.org/documents.aspx
For commercial use, the UAV currently must be certified like a manned aircraft, which is impossible in practice for small UAVs. Until recently, this has rendered the commercial exploitation of UAVs impossible in US. The parliament has been tasked to design laws to permit commercial exploitation of UAVs by 2015 at the latest. At the moment, it's possible for government agencies and university to obtain special permits through Certificates of Authorization. Other ways of demoing/using UAS are also being put in place (testing fields, lightweight certification process, etc.), but it will be difficult to have a clear picture for the next year or two. To complicate the picture, several states are in the process of passing restrictive laws related, e.g., to privacy, so the situation may end up being state dependent.
It is best to check with your local council - they will be able to inform you on general laws as well as local by-laws. Another good source would be your local model plane club.
As far as I know: UAV typically requires permission, depending on capabilities (they don't want unlicensed Predator drones flying around). Quadcopters typically don't, except at public events (like those camera quadcopters they use at some sport events).
Disclaimer : based on my personal understanding never having been to the states (they are pretty much standard the world over). I accept absolutely no reponsibility for any claims arising from usage of this info i.e. use at your own risk.
You can fly anything you want in class G airspace. Generally speaking, if you're far from any airport or military training area, you are probably walking around in class G. To be sure, though, you should consult a current FAA sectional chart for your area.
Class G is considered "uncontrolled" by the FAA. Note however that there are other regulations and agencies that restrict flight activities there. Sadly there is no "easy" answer in U.S. airspace.
In most cases, except for special situations like in the mountains, Class G extends up to 1200 ft above the ground (AGL).
I'm quite familiar with airspace rules because of my previous job (UAV operator). I guarantee you can fly anything you want, but see for yourself: http://en.wikipedia.org/wiki/Airspace_class_(United_States)#Class_G