In Skyrim and on Solstheim, aurora may randomly appear in the night sky when it is clear. The aurora come in a variety of different palettes of color and will last for some time. It serves no purpose in The Elder Scrolls V: Skyrim other than aesthetics.
If the Clear Skies shout is used when the aurora can be seen, the aurora might disappear. After some time, it might come back, (sometimes in a different color). Sometimes the aurora does not come back at all after using the shout.
Alternatively, the Clear Skies shout can be used at night when clouds are obscuring the aurora from view. When used, the clouds will disappear and the aurora will be seen with no trouble at all.