Why do the Arctics have days that are 24 hours long?
Of course, the polar regions do not always have days that are 24 hours long (I assume you are talking about times when there is sunlight all around the clock). But when it is summer time in their particular hemisphere, these regions do have times when the sun does not go down.
This is because of the fact that the polar regions point more directly toward the sun during the summer months. Because they are pointed so directly at the sun, the Earth's rotation cannot cause the sun to "set." No matter how the Earth rotates, the pole is pointed straight at the sun and so the region gets sunlight 24 hours per day.
Arctic regions have day and nights that are not just 24 hours long, but much more than that. These regions have a six month of what may be described as day followed by six months of what may be described as nights.
The cycles of days and nights is caused by rotation of earth along a north south axis. This causes different pars of the earth to face the sun during one complete rotation of earth which takes about 24 hours to complete. The relative direction of axis of rotation of earth and the orbit of earth around the sun are arranged in such a way that as the earth orbits round the sun the duration of for which different parts of the earth face the sun during the one rotation of earth varies from place to place on earth, and from time to time during one complete orbit round the sun.
The axis of rotation of earth is tilted at an angle of 23.5 degrees with respect to the plane of rotation of earth round the sun. As a result the area in the arctic region which falls beyond 66.5 (90 - 23.5) degrees north and south and longitude remain exposes to or hidden from the sun for continuous period of six months, result in days and nights of six months.
Because of tilt of axis of earth, at one time during its orbit round the sun - around January 14 every year - the north pole is tilted maximum towards the sun, while the south pole is tilted maximum away from the sun. As a result the the area around the South Pole pole up to 66.5 degrees south is exposed to the sun for the complete rotation of earth resulting in day for the entire 24 hour period of rotation. Similarly, area around north pole up to 66.5 degree north is hidden from the sun resulting in night for the entire 24 hour period of rotation.
The relative position of the North and South Pole with respect to sun gets reversed when the earth comes to the exactly opposite of its orbit after completing half of its orbit. The the North Pole faces the sun for entire 24 hour rotation of the earth while the South Pole remains hidden from the sun.