Alcoholic beverages are very expensive in Copenhagen. A 500ml beer ( 0.87 Imperial Pint, 16 US fluid ounces) served in a pub averages about 48 Danish Kroner (approx. 6.45 Euros, 5.60 Pounds Sterling, 9.05 US Dollars), but some charge only 20-30 Kr [I'd like to know where!], especially on weekdays or during early hours, while fancy places obviously charge more. Unless you come from elsewhere in Scandinavia don't frighten yourself by trying to work out what this costs in your home currency. At most places the beer on tap is either Carlsberg or Tuborg. In either case there will be a choice of the normal pilsner and then a slightly redder special or classic. Some might also offer wheat or dark beer.
If you are on a budget you could follow the example of local teenagers and get primed with bottled beer from a supermarket or kiosk (3-7 Kr for a 330 ml bottle). It is legal and very popular to drink beer in public (not on public transport, although it will be tolerated if you are not showing drunk behaviour), so buy a beer, sit on a park bench or at Nyhavn and enjoy Danish life.
As for where to drink, most tourists head straight for Nyhavn but while indeed pretty, the high prices here make it a bit of a tourist trap. In good weather imitate the locals by buying beer from a kiosk and dangling your legs over the water or head elsewhere to get your drinking on. The many side streets north and south of the strĝget pedestrian street are a good starting point. Other good areas are Vesterbro west of the central station, along Vesterbrogade and Istedgade and in the meatpacking district. On Nĝrrebro, the cluster of bars and clubs around Sankt Hans Torv and Blċgċrds Plads, just after the lakes, is another hotspot. For a coastal city Copenhagen has surprisingly few places where you can enjoy a water view with your beer or coffee.
Charlies, Pilestrĉde 33, ? +45 33 32 22 89, . M-W 2PM-2AM,Th-Sa noon-2AM,Su 2PM-9PM. Great choice of beers, especially British beers and real cask ales. Very popular so you might have to stand on weekends, but lively also on weekdays. edit
Cosy Bar, Studiestrĉde 24, ? +45 33 12 74 27. Open every day of the week. A small late night bar/club with a very mixed clientele, extremely busy in the weekends, but the fun usually never starts until 4-5AM when other places are closed. Fun, but can feel a bit like a meat market at times. edit
Den Tatoverede Enke, Gothersgade 8C, ? +45 33 91 88 77 M-W 4PM-midnight,Tu-Th 2PM-1AM,F-Sa 2PM-2AM.. The name means the Tatooed widow in Danish, 16 beers on tap and more than 100 on bottles. This is the place for Belgian beers. Also have a good restaurant on the top floor. edit
The Dubliner, Amagertorv 5, ? +45 33 32 22 26. M-Th 10AM-late,F-Sa 10AM-5AM,Su 11AM-late. Huge and rather touristy, it is still probably the most authentic Irish pub in Copenhagen, usually always something going on, even on weekdays - also serves pub food, and has many big screens for watching sports. edit