There is no unique definition outlining an “Arctic region”. In fact, the parameter or the highlighted characteristic chosen will outline different shapes, including or excluding some areas. Likewise, the is not one correct definition, as the choice much depends on the topic addressed.
Here you can take a look to the main Arctic definitions currently in use (click on the map to make it larger):
The Arctic Circle includes an area north of the Arctic Circle at 66° 33’, that is the Southernmost latitude in the Northern Hemisphere at which the sun can continuously remain above or below the horizon for 24 hours.
As seen by the map above, the area so defined include only the Northernmost island of Iceland, Grimsey, most of Greenland, Nunavut and part of the Northern territories, the northernmost part of Alaska and Russia, and Lapland.
The tree line is the northernmost border where trees can grow
10° July Isotherm
The average temperature in July of the area described by the 10° July isotherm line never goes above 10° Celsius.
Arctic Human Development Report Bounders
“[AHDR] have concluded that there is much to be said for bounding the Arctic in a manner that is broadly compatible with studies of other Arctic issues rather than adopting yet another approach to determining the extent of the region. For this reason, the AHDR takes as its point of departure the region that the Arctic Monitoring and Assessment Programme covers in its 1997 and 2002 reports (4-5). For reasons having to do mainly with the location of jurisdictional or administrative boundaries and the availability of data, however, the area covered by this report differs from the AMAP Arctic in some respects. Thus, the AHDR Arctic encompasses all of Alaska, Canada North of 60°N together with northern Quebec and Labrador, all of Greenland, the Faroe Islands, and Iceland, and the northernmost counties of Norway, Sweden and Finland. The situation in Russia is harder to describe in simple terms. The area included, as Introduction: Human Development in the Arctic 17 Chapter 1 demarcated by our demographers, encompasses the Murmansk Oblast, the Nenets, YamaloNenets, Taimyr, and Chukotka autonomous okrugs, Vorkuta City in the Komi Republic, Norilsk and Igsrka in Krasnoyarsky Kray, and those parts of the Sakha Republic whose boundaries lie closest to the Arctic Circle”
"Given the different definitions of the Arctic, based on physical-geographical characteristics as described above, and those based on political and administrative considerations within different countries, no simple delineation of the Arctic region was applicable for the purposes of the AMAP assessment. To establish a geographical context for the AMAP assessment, therefore, a regional extent was defined based on a compromise among various definitions. This incorporates elements of the Arctic Circle, political boundaries, vegetation boundaries, permafrost limits, and major oceanographic features. The region covered by AMAP is, therefore, essentially the terrestrial and marine areas north of the Arctic Circle (66°32'N), and north of 62°N in Asia and 60°N in North America, modified to include the marine areas north of the Aleutian chain, Hudson Bay, and parts of the North Atlantic Ocean including the Labrador Sea. As stated above, the AMAP boundary was established to provide a geographical context for the assessment, in particular source-related assessment issues, i.e., consideration of sources within and outside the Arctic. The relevance of the AMAP boundary varies when considering different issues, and it has therefore been applied accordingly. Thus, contaminant levels in biota are addressed in relation to the geographical occurrence of the species concerned; demographic data are discussed in relation to administrative regions on which, for example, census data are collected.
The "Agreement On Cooperation On Aeronautical And Maritime Search And Rescue In The Arctic” defines the “Scope of Application of this Agreement” in the Annex, outlying a different shape for the Arctic region.
The areas is delimited by country, by continuous lines connecting specific coordinates listed in the Annex itself (see here).