Old timey charm and
lots of shops
Why locals love it
These several streets are usually all bunched into this one name, Strøget. It's a pedestrian zone (no biking either!) that stretches through a large part of the central city. The side streets are usually bunched in with this, too, and make for an essential Copenhagen stroll.
Why you should visit it
Well, even though most of it is shops, cafes and such, the downtown area is very charming with houses dating back to the 1600's and earlier. You'll get a feel for Copenhagen as well as walk by several tourist attractions. Start at the Stork Fountain near Christiansborg and explore.
The side streets are a good bet for smaller shops with more interesting fare than the chain stores on the main streets.