They speak several languages of their own from the Finno-Ugric family, related to Finnish but not to Swedish and kin. See Saami language.
See also aborigine?, other Northern peoples: Eskimo, Innu?, Inuit, Inuvialuit?, Aleut?.
For various Russian peoples, see http://www.geocities.com/Athens/Oracle/8226/sibnative.html (They list 85 groups. I don't know which ones are Northern.)
The Nordic FAQ at http://www.lysator.liu.se/nordic/scn/faq23.html has a section on the Sámi.