The currency is the Norwegian krone NOK, and this is also accepted in the Russian settlements. Svalbard is a tax free zone so a number of shops in Longyearbyen display various items for sale at prices well below mainland Norway's.
costs
Svalbard is by most measures horribly expensive: mainland Norway is bad enough, but on Svalbard everything costs even more. Accommodation in cheap guesthouses costs on the order of 500 kr/night and sit-down meals nudge up closer to 100 kr each - both figures you can very easily double if you want to stay in a full-service hotel. Guided activities start at about 500kr per day e.g. trekking and kayaking but can go to 1000kr and above for tours requiring specialist equipment.
One way to cut costs significantly is to camp and self-cater, bringing all your supplies from the mainland. There is, however, a full service grocery store in Longyearbyen. Frozen and dry goods are on par with or even a little cheaper than in Norway, while perishable items arrive via air freight and are more expensive.
Svalbard's duty-free status means that alcohol and sports clothing, etc. are actually much cheaper than on the mainland.