There are different options on where to live, if you come to work or study in Denmark. If you are going to study, it is most common to live in student halls. Danish colleges and universities do not have on-campus housing. It is however, not very easy to find a room, especially in the larger cities like Copenhagen and Aarhus.
Other options are apartments. You can live by your own or you can share an apartment with other young people. This makes the rent cheaper, so it is quite popular.
This page might help you: http://www.ungdomsboliger.dk/?id=navigationskort&lang=en
Or this one: http://translate.google.com/translate?sl=da&tl=en&js=n&prev=_t&hl=da&ie=UTF-8&eotf=1&u=http%3A%2F%2Fstudenterguiden.dk%2Fstudieboliger-kollegier&act=url
Prices are very different.
Student halls, all over the country, are at a range from Euro 330 – 730 per month (the most expensive have 2 rooms)
An apartment in Copenhagen ranges from Euro 650 – 1100 (most expensive have 2 rooms)