I live in Copenhagen, so I have never had the need to use Airbnb. It is easy to get from the airport to the city, either by metro or regular train. Use http://www.rejseplanen.dk/ to plan your journey once you have an address.
Which neighbourhood to stay in depends on you and your interests, so i'll just give you a highlight of the different neighbourhoods.
The city center is the old part of the city, it's where most of the classic sights are, but it is also a bit touristy with lots of bars in some areas. Also it is not really a residential area, so there might be limited Airbnb apartments available and they will be more expensive.
Vesterbro is an old working class neighbourhood but has seen some gentrification in the past few years. Today there are many students and hipsters living there and lots of cool bars and clubs, especially in the old meat packing district. It also has a small red light district (though prostitution is illegal) and the area around Istedgade close to the central station is a bit seedy, but not dangerous.
Nørrebro is also an old working class neigbourhood with a large immigrant and student population, which gives it a lot of atmosphere and cheap restaurants, especially around Blågårdsgade and Jægersborggade. It has also had a lot of bad press for many years with crime and gangs, but don't believe the bad hype. I live here and i am not worried about going out at night or any of that stuff.
Østerbro is a posh, upper class neighbourhood with lots of cafés.
Frederiksberg is not really a part of the City of Copenhagen, but an enclave within. Its posh and upper class with rather grand buildings and nice cafés.
Christianshavn is an old sailor neighbourhood inspired by Amsterdam, with canals and narrow houses. Very scenic with a nice atmosphere, but could be expensive. it also includes the Freetown Christiania, an old hippie comune well worth a visit.