Elysium has an axial tilt of 88
What does that look like?
Earth has a 23 degree axial tilt, which means that at midsummer/midwinter the poles are tilted 23 degrees towards or away from the sun. This tilt gives us our seasons.
Near the equator, day and night are roughly the same length no matter what time of year it is. But the further away from the equator you go, the greater the difference between summer and winter hours of daylight. When you cross the Arctic or Antarctic circles - lines of latitude 23 degrees from the poles - something strange happens. You get periods of round-the-clock daylight in summer and darkness in winter. The closer to the poles you travel, the longer those spells of continuous light and dark last. The poles themselves see close to six months of light and six months of dark.
On Elysium, this picture is taken to extremes. The Arctic and Antarctic circles would actually run along the equator, and everywhere other than the equator will be a “land of the midnight sun”, even if only briefly at the lower latitudes.
Because the planet’s spin is in line with the plane of its orbit, the motion of the sun in the sky will look very strange compared with what we are used to. At midsummer, the planet’s axis is pointing right at the sun, so the sun will be stationary in the sky - directly overhead at the summer pole, or hovering on the horizon if you’re at the equator. This will make for ferociously long and hot summers at the poles.
As the year progresses, the sun will start to move in increasingly large circles in the sky, a bit like the handle of a spinning top that’s losing speed and starting to topple over. Sooner or later, depending on what latitude you’re at, those circles will dip below the horizon and you’ll get a few weeks or months of a true day/night cycle ... until the sun vanishes below the horizon for good and you go into a much longer night.
Why doesn’t one of Elysium’s poles always point at the sun?
That’s a popular misconception and an easy mistake to make.
Yes, some worlds do have one side always facing their sun, but it’s not their pole. Their axis is the more normal perpendicular (or near enough) to the plane of the orbit like in this first picture.
The perpetual day/perpetual night happens when the planet’s day (one rotation on its axis) is exactly the same length as its year (one full orbit of its sun).
This may sound like one heck of a coincidence, but tidal forces between two orbiting bodies act as a braking mechanism, slowly bringing rotational and orbital periods into line. This is called “tidal locking”. A familiar example of this phenomenon is our own moon, which always shows the same face to the Earth. For many years, astronomers thought Mercury was tidally locked to the sun, although that’s now known not to be true.
Elysium’s axis is not even close to perpendicular. It’s lying on its side like in the second picture here.
If you were watching it from somewhere outside the solar system you’d see that the planet’s axis is always pointing the same way relative to the universe as a whole. This means that each pole gets a dose of summer sun directly overhead, followed by a long winter where it’s pointing away from the sun.
Again, this sounds bizarre and far-fetched, but our very own Uranus is doing exactly this.