It is midnight sun, because the sun is up at midnight.

However, there is no "night of the midnight sun" because, quite simply, there is no night. It is simply daylight all the time.

At exactly midnight, on the arctic circle, on the summer solstice, the sun will be on the horizon, similar to sunset, except that it will never dip completely under the horizon - it will just drop down, kiss the horizon, then come back up again. The colors will be similar to those at sunset. As with anyplace on earth at sunset, you cannot see stars, and only the brightest planets would be visible. Auroras would not be visible, as they would be overpowered by the skyshine.

Here is EXACTLY what it looks like (that was taken at right around midnight in northern Europe). Yes, I know you don't want links, but that one picture answers more than half of your questions far more effectively than words ever could.