The Central Station area is the first place most folks—business visitors and tourists alike—go to find a hotel. Trains to and from the airport can be caught at the Central Station, or Hovedbanegarden. Additionally, the area is home to many of Copenhagen's main attractions.
Famed Tivoli, the Ny Carlsberg Glyptotek museum, Radhusplasden (aka Town Hall Square) and the Stroget shopping streets are all clustered near the station. Though the area is quite safe, it can seem dodgy at night, when it morphs into Copenhagen's Red Light District.
Frederiksberg is a lovely neighborhood situated west of the Central Station (and Tivoli) area. Legally a town in its own right, Frederiksberg has a governor, a population of 91,000 and tree-lined residential streets.
Most famous for Frederiksberg Have, a spacious, serene park filled with trees and canals, the area also boasts a number of other tourist sights. The Royal Copenhagen Porcelain Manufactory, the Carlsberg Museum and the Carlsberg Visitors Center are all here.
Picturesque Nyhavn Canal, with its Easter egg-colored houses and tall ships, arm-wrestles with The Little Mermaid statue for Copenhagen’s most photographed tourist site status. The good news is that, despite the legions of canal-tour boats pouring in and out of this narrow strip of water, Nyhavn and its numerous outdoor cafes are still a lovely (though somewhat pricey), place to soak in old-school Copenhagen ambience while feasting on smorrebrod, an open-faced sandwich.
The canal sits just off of Kongrens Nytorv, an enormous square that anchors the end of the Stroget shopping area.
Old Town is the geographic center of Copenhagen. The area's primary claim to fame is the Stroget, Europe's longest pedestrian shopping street. (Don't look for it on a map as it's made up of five different streets, and none share its name.) Anchored on the west by Raadhuspadsen with a host of souvenir shops, Stroget slopes east to Kongens Nytorv.
In addition to a smorgasbord of shopping, Old Town is also home to a student friendly and plaza packed Latin Quarter and to the historical and political center of the city, Slotsholmen. Oh, and to hotels. This being the fashionable district and the city's center (and expensive Copehagen), most hotels of Old Town are not what we'd call cheap, but for those who insist on being in the center of things, there are deals to be had.