In the US they are not used very commonly around airports. You are correct that they are used for bird or other wildlife control and monitoring. They have also been proposed to be used to provide security monitoring. More appropriately, an unmanned vehicle may be better suited for runway, taxiway, and apron inspection which is not done by manned vehicles. We have conducted a significant amount of research looking into using UAS to assist airport rescue and fire fighting.
I've completed my postgraduate degree in the University of Southampton, during which time I've lived near the Southampton airport.
From what I know UAVs and any kind of flying toys / drones are banned within a considerably large radius from the airport and the commercial flying zones. Check this out: http://www.noflydrones.co.uk/
You'll find that drones of almost any kind, except for under direct control by the government, the military and/or authorized therein - even in the case of airports, are not allowed to fly in these airspace. A huge part of the UK's airspace is strictly no-fly zones for all drones. I'd imagine it's the same case in many other countries as well.
Drones usually are too small to appear in normal commercial aircraft radar and too difficult to control in current day technology and infrastructure, except for to ban them altogether. Like David Carl Ison says, they are used for bird or other wildlife control and monitoring; however from the best of my knowledge these kind of applications are in wildlife scantuaries and national parks. As far as airports are concerned, I don't think getting permits to use drones to ward off birds will be too easy. Not to mention the legal consequences of animal rights and the safety / insurance regulations for the rest of the people and property that is not only involved but also in path of the possible failure of the drone.
While the common use is still somewhat limited due to mentioned regulatory concerns, possible additional (to those already mentioned) future uses could include: aircraft inspection (e.g. from the ground hard to access areas such as top of fuselage and vertical fin), infrastructure inspection and security (to include remote sites such as NAVAIDs, approach lighting systems, antennae structures), and flight verification tasks for navigation systems (e.g. testing ILS or approach radar accuracy)...
I'm pretty sure there is more to come as the technology and its civil application is rapidly evolving. Besides pure UAS/UAV uses, it's probably also worth mentioning other possible unmanned/robotic solutions around airports (e.g. self-driving luggage carts)...
Of course in Poland there are a lot of restrictions - but airport operator have right to use drones (after receiving approval from services / after contact with tower ). For spotters these restrictions are even higher.
Recently in Poland one of airport operators used/tested drone to monitoring area surrounding the airport (to protect infrastructure ) and to detect other drone.
Most countries (developed) have some kind of legal guidelines on where drones can and cannot be used. I believe one common thread is that they should not be operated near aircraft or airports, with some regulations specifically prohibiting use within a certain distance of airports. So aircraft spotters should generally NOT be flying their drone near airports.
As far as I know some airports have already started to use drones for inspection of landside elements such as, taxiways and runways, buildings and perimeter fence. Although there are so many regulation requirements regarding drones, they have a great potential for airport operations too.
The airports that use drones for their operations often use tethered systems to ensure the safety. This way, there is no real risk of intervening with the conventional flight operations and the drones can be used for all kinds of tasks (maintenance, inspection, ...)
The airlines sometimes use drones to keep away the birds from airplanes. I know the FESTO Company designed and manufactured a flapping wing similar to sea bird to scare the smaller birds,