Due to the 23.5 degree tilt of the Earth, the northern hemisphere is tilted toward the Sun in the summer and away from the Sun in the winter. Inside the Arctic Circle, the Sun will spend a period of days completely above the horizon during the summer months and completely below the horizon during the winter months. The number of days depends on how far into the Arctic Circle one goes. (Same is true for Antarctic Circle in the Southern Hemisphere, but at opposite times of the year.) The closer you are to the poles, the greater the number of daylight hours. For example, from March 21 to September 21, the Sun is always above the horizon if you are standing on the North Pole.
Although the Sun does set on all days over most of Iceland, Iceland's proximity to the Arctic Circle means that the number of daylight hours varies dramatically over the year. Below is a chart showing this.
We visited Iceland in mid-July when there were about 20 hours of daylight. Even when the Sun set, it didn't drop far enough below the horizon to get dark. The skies darkened a little big, but it was still essentially light out. However, if you visit in December, you may only have 4-5 hours of daylight. That's not much daylight!!!
Visiting Iceland in the summer was awesome because we never worried about it getting too dark to do something. Visiting a water fall at 1 AM? No problem, it's still light out!!!