The best time to see the Northern Lights from anywhere around or above the Arctic Circle (which lies near the towns of Rovaniemi and Bodø) is any time between September and late April. You'll experience long winter nights here.
Further south, the season of the Northern Light is shorter, partly because there is more light in the months before and after winter. Between mid-October and March is the best time to see the Northern Lights in that region.
The optimal time of night for the Northern Lights is 11 p.m. to 2 a.m. Keep in mind that most visitors head out to begin their watch around 10 p.m. and conclude their night around 4 a.m., since the Northern Lights can be hard to predict (just like the weather in Scandinavia).
Don't forget to learn about photographing the Northern Lights if you're planning to bring your camera.
If you do not see the Northern Lights as expected even if the timing is right, locals recommend to simply wait 1-2 hours. Nature tends to reward the most patient.
Get the current forecast for Northern Lights in Part 4.