Accommodation in Greenland tend to be pricey. World class hotels exist in all of the more visited areas Hotel Hans Egede in Nuuk, Hotel Arctic - with its igloo rooms - and Hotel Hvide Falke in Ilulissat, but cheaper options exist. Try for the Seaman's Home hotel in Maniitsoq, Nuuk, Qaqortoq, Sisimiut and Aasiaat.
For less expensive options, you can check with the Nuuk Tourism office for its hostel program, where locals have rooms they will rent out for a third the price of the town's hotels. It's a great way to experience the real Greenland, although knowing a few words of Danish or Greenlandic is very helpful as your hosts may or may not understand English. Additionally, since there is no private property ownership in Greenland you can also camp in any field or plain for free if you're equipped to handle the elements.