In Europe, the main areas to see the Aurora Borealis or Northern Lights are in the northern parts of Sweden, Norway, Iceland and Finland (or most famously, Lapland, which is a part of Finland). You can also see them in Scotland and parts of northwest Russia. They have even been seen in England on occasion. Southern and central areas of the Scandinavian countries may see the Northern Lights sometimes, but not as often or as clearly as the northern parts.
In America, the lights can also be seen in parts of the US, especially Alaska, and Canada.
The best locations from which to see the lights can change somewhat, because they are centred around the north magnetic pole, and that slightly changes from year to year. There are a number of companies that specialise in tours to the Aurora Borealis regions, and these change their offers depending on where the best views are to be found. You can find one example here.