Why is the sky of the moon always dark?
The moon does have a night and a day, but this isn't as fully connected to your question as you might think.
The moon is tidally locked with the earth, meaning that the same side always faces earth. Since the moon also orbits around the earth (with a period of a lunar month), this means each side changes, over the course of a lunar month, between facing towards the sun and facing away. This is the cause of the phases of the moon, and also describes the day/night cycle for the moon: a full moon is when the side that we can see faces the sun - i.e. "day" for that side - and a new moon is when the side that we can see faces away from the sun - "night". So the moon does have night and day, but a night/day cycle is one lunar month long.
However, this doesn't answer your question of why the sky is always dark when viewed from the moon - even when the sun is above the horizon.
The reason that the sky appears bright on Earth is that, even when there are no clouds, the atmosphere scatters the sunlight - meaning that only about 75% of the sunlight that reaches the ground appears to come from the sun, while the rest comes from all over the bright sky. This is called Rayleigh Scattering, and is the reason that the sky appears blue.
The moon has no appreciable atmosphere to do this scattering, so the sky appears dark.
With the atmosphere of the moon being $10^{14}$ times less dense than that of Earth, there is negligible scattering, so whereas on Earth, approximately 25% of direct solar radiation is scattered around (making the sky light up and appear blue), there is no mechanism for this on the moon, and all light from the sun travels (essentially) unaffected to the surface.
In addition to the lack of an atmosphere (the primary reason, already addressed in other answers), something that compounds the effect even further is that the lunar surface is quite reflective, and because the people who are taking the images of lunar surfaces often want to have Earth and/or astronauts in the shots and also be able to make out detail in the shot and an illuminated surface, it is reflecting light from both the sun and the Earth.
The relative brightness of the surface causes the camera's image sensors / aperture settings to need to restrict the amount of sensitivity (ISO) or length of exposure (aperture), this make the already atmosphere-less moon appear to be surrounded in darkness when part of what you are seeing is compensation for the rather bright surface in relation to what we are accustomed to on Earth.
Note: I searched for a while to find an image from the lunar surface at 'night', where presumably you would be able to see stars in the background, but I was unable to find any. Do we not have any pictures like this? Is there a limitation due to temperatures and the cameras that prevents taking them perhaps?