In our Solar System, do the moons of other planets have night and day like Earth's moon does?
'Regular moons' are natural satellites of planets that are relatively close to the planets they orbit, have a prograde orbit (they orbit in the same direction as the planet they orbit), have little orbital inclination (their tilt relative to the planet they orbit is small) and have low eccentricity in their orbit path (the orbit is even about the planet, that is more circular than elliptical). It is believed they form in orbit about their primary (the planet they orbit) unlike 'irregular moons' that are believed to have been captured whilst passing near the planet.
There are at least 55 regular satellites of the eight planets: one at Earth, eight at Jupiter, 22 named regular moons at Saturn, 18 known at Uranus, and 6 small regular moons at Neptune.
All but one of the regular moons in our solar system are 'tidally locked' to their primaries, that is they rotate in synchrony such that the same side of the moon is always facing the planet it orbits. This is the case with our moon here on Earth, so that we only see one face of our moon. The only known exception is Saturn's moon Hyperion which rotates chaotically because of gravitational influence from another moon of Saturn: Titan.
The phases of the moon that we see here on Earth are the phases of its rotation about its own axis, that is, the moon's day and night. Our moon takes about 28.5 days to rotate about its own axis, which is then one moon day ('day' refers to 'day+night') - as the moon spins about its own axis, when a particular point on the moon is facing away from the Sun it is moon-night there and when it faces towards the Sun it it is moon-day there. The length of a moon day is roughly the length of a month. The moon has a big effect on the seasons, through its effect on the oceans by gravity, creating tides, which is why we divide our year up roughly according to the moon's rotation, ie months. We always see the same face of the moon when we look at it from Earth (it is 'tidally locked'), since the time it takes the moon to orbit the Earth is roughly the same length as a moon day. This is mainly true of all the regular moons in our solar system, except Saturn's Hyperion as mentioned above.
For example, Jupiter's closest moon, but fourth largest, Io, takes 42.5 hours to rotate about its own axis. This means that Io's day (day+night) is 42.5 hours long. As with all other bodies in the solar system, the angle with which the moon is pointing towards the Sun at any one time determines how much of this day is daytime and how much is night-time for a particular place on the moon. If the moon points towards the Sun (as in Summer here on Earth), more of the day is daytime for places on the moon nearest the Sun, and if it points away from the Sun (as in Winter here on Earth) more of the day is night-time for places on the moon furthest away from the Sun.
A regular moon that is known not to be 'tidally locked' is Saturn's Hyperion, which has a chaotic orbit - the only moon in the solar system to have such an orbit. It is oddly shaped, and not round as many moons are found to be, and is close to another of Saturn's moons, Titan, whose gravitational pull may affect the regularity of Hyperion's orbit. By 'chaotic orbit' it is meant that 'its axis of rotation wobbles so much that its orientation in space is unpredictable'. This means that its rotation around its own axis is less stable, meaning that its length of day is less stable. Because of this, although Hyperion will have a day and night like the other moons in the solar system, the length of these may be less predictable from time to time than other moons. The orbital period is 21 days, but the rotational period (about its own axis) is not in synchrony with this (not 21 days) and is quoted as 'chaotic'.
Most moons will experience day and night just like the Earth's moon. Moons rotate just like planets do, so the sun will hit different parts of the moon at different times, creating a night and day. This is also why we have moon phases, where we only see sections of the moon's face and the rest is dark; the moon has rotated causing the sun to hit a different place.