This is really difficult, but I think there might be one or two reasonably plausible natural situations, and at least one artificial situation created by an advanced civilization, where one part of a world has alternations of light and dark and another part has only eternal darkness.
To appreciate some of the problems with such requests, you might want to look at an attempt to design a vaguely similar situation in these two posts:
PART ONE: A TIDALLY LOCKED PLANET
All planets and other astronomical bodies rotate at various speeds determined by various factors.
So if a planet orbits around a star close enough to get enough light and heat from that star to be warm enough for life, presumably like Earth life, it will have days and nights. At any one moment half the surface of the planet will be receiving light from the star and the other half will be in shadow and dark. As the planet rotates every part of the planet's surface will alternately experience day and night.
If the axis of the planet is tilted the relative length of night and day will vary with the seasons at different latitudes. But even at the poles periods of constant light or constant night must last less than half of the planet's year. So no spot on that planet can have either eternal light or eternal night.
Except that the distance from the planet to the star where the planet can have the right temperature for life varies with the luminosity of the star, and the luminosities of main sequence stars vary according to their mass. And a small change in the mass of the star will cause a much larger change in the luminosity of the star.
So a small mass star will be really dim compared to the Sun, and thus the habitable zone around that star will be really close to it. Any planets orbiting the habitable zone around a low mass, dim star will orbit really close to it. And thus the tidal forces on the planet from that star will be very strong and the planet will become tidally locked to the star very soon in geological time.
As the planet becomes tidally locked to its star, its rotation rate will be slowed and slowed until the time it takes for the planet to make one rotation will be the same as the time it takes for the planet to orbit around the star one time. Thus one side of the planet will always face the star and will have eternal day, and the other side will always face away from the star and will have eternal night.
So how do you get the side facing the star to have alternating day and night, while the other side has only eternal night?
That is very tricky.
PART TWO: a TIDALLY LOCKED MOON, FIRST POSSIBLE SOLUTION
The original question asks for a moon orbiting a planet, with the moon being tidally locked to the star that the planet and moon both orbit. But calculations indicate a moon would have to be tidally locked to its planet instead of to its star. it is considered impossible for a moon to be tidally locked to its star instead of to its planet.
One could try making the planet a planet sized moon orbiting a giant planet and tidally locked to the planet, and not to the star. Then the side of the moon that always faced away from the planet would have alternate day and night. The side that always face toward the planet should also have alternate day and night, except that its day should be interrupted by a long eclipse every day when the moon passes into the shadow of the giant planet. And if the giant planet has a bunch of other large moons, the moon in question might often be in eclipses caused by them.
So I suppose that it is theoretically possible that someone could design a plausible configuration where at at least a small segment of the side of the moon that always faces the planet would always be in eternal darkness.
PART THREE: ECLIPSING BINARY, SECOND POSSIBLE SOLUTION
Another possibility is a planet orbiting in the habitable zone close to a dim star, and tidally locked to that star, as discussed above. Except in this case the star is a close binary, a very close binary, and the planet orbits around both of the stars.
Astronomers have theorized about planets in such orbits, called P-type or circumbinary orbits, and have detected some. It is possible that two really dim stars could orbit close enough to each other that a planet in their combined habitable zone would orbit close enough to be tidally locked to the stars. Thus one side of the planet would have eternal night. So what about day and night on the side facing the two stars?
The planet's orbit would probably be dragged by tidal forces to orbit the two stars in the same plane as they orbited around each other. So each of the two stars would periodically eclipse the other one as seen from the planet - they would be an eclipsing binary as seen from the planet.
And if one of the stars is a vary dim star, or less than a star, it would contribute very little light to the planet, and when it eclipsed the bright star a sort of night would fall on the star facing side of the planet. So the dimmer object should be cool enough to emit very little visible light, and also large enough in diameter to totally block off the light of the star when eclipsing it.
So the dimmer object should be a "puffy planet" or a brown dwarf.
And I guess that various possible configurations of such a system might have eclipses on the star facing side of the planet last for about 0.1 percent of the time, or 1 percent of the time, or 10 percent of the time, etc., etc. That would not be like having equal periods of day and night, but it would be a lot different from having eternal day on that side of the planet.
PART FOUR: A ROGUE PLANET WITH AN ARTIFICIAL "SUN" SATELLITE, THIRD SOLUTION
A rogue planet is a planet in interstellar space instead of in a star system.
We can expect rogue planets far from any star to be very, very, very cold. But an advanced civilization could terraform a Earth-sized rogue planet and make it habitable. There would be many possible methods to heat it to the desired temperature.
One such method would be to create an imitation of the geocentric model of the solar system, with a "sun" orbiting the planet instead of the planet orbiting the "sun". The advanced civilization would build a gigantic artificial sun" satellite to orbit the planet, with countless gigantic fusion generators to power countess gigantic lamps aimed at the planet.
The artificial "sun" satellite would orbit the rogue planet, creating a repeating "sunrise", day, "sunset", and night sequence over every part of the planet.
But if for some reason the highly advanced civilization wants to create a world with eternal night on one side, and eternal day on the other side, they can put the artificial "sun" satellite in a synchronous orbit so it is always above one place on the planet. So one side of the planet will have eternal day and the other side of the planet will have eternal night.
But if for some reason the highly advanced civilization wants to create a world with eternal night on one side, and alternating day and night on the other side, they can put the artificial "sun" satellite in a synchronous orbit so it is always above one place on the planet, and then periodically turn the artificial "sun" satellite on and off. So one side of the planet will have alternating day and night and the other side of the planet will have eternal night, as requested.