Nørreport is an area which has gone from quite "grey and boring" to much more interesting and "almost hip". Right now as we speak (or write), even the area around the station is having a big make over (causing a lot of troubles and frustration among Copenhageners who need to pass this spot in town on their way to work every morning).
The old discount shopping centre Daells Varehus is in many ways a symbol of how Nørreport has changed. For several decades this shopping centre was probably the most important thing in the area (at least in terms of shopping), and even though I personally think about Daells Varehus with a bit of nostalgia because I have so many memories of my mum taking me here to buy clothes, it honestly wasn't the most exciting place to shop, just like the rest of the area.
Today Daells Varehus has been turned into one of the most hip hotels of Copenhagen, Skt Petri, and the street, Fiolstræde, which used to house shops within the same discount category, now has some quite interesting places in terms of cafes, clothes, shoes and children's clothes shops.
On the other side of Nørreport you find the new indoor market Torvehallerne, and then you mustn't forget that a part of the University of Copenhagen is located here, so the area is also filled with students, Danish as well as foreign exchange students.
The most important street around Nørreport is probably Købmagergade which starts from the station and goes all the way down to the main shopping street, Strøget. Also Købmagergade has undergone a big change within the last years and now you find many really great places here.
Nørreport is also where you find some of the best green spots in Copenhagen. Here is the Botanical garden, the entrance to Kongens Have and you have the quite lovely park, Ørstedsparken. These are all great places, when you want fresh air on a warm summer day (Not that there are many of them!).
So even though Nørreport continues to be kind of a transit area in Copenhagen (you usually pass Nørreport when going to or from another place in town), it has definitely become much more interesting now than it used to be.