Why is night vision lacking color? When I look through night vision with a video camera or see it in movies it always seems to be sort of grey or white. Why doesn't night vision have color?
 A: "Night vision", in the sense of a device to help humans see in the dark, can refer to different things. The most commons are:
1. Residual light enhancement
These rely generally on the principle of image intensification in which incoming light is converted into electric charge via the photoelectric effect and then amplified with a microchannel plate detector (in red) via an external source of energy (often a battery). The process is roughly depicted in the following graph. The "residual" light hits a photocathode (in gray) and the emitted electrons are accelerated towards a phosphorescent plate (in green) that emits light in response. In this process the amount of electrons initially collected is increased up to a thousandfold which is how the camera can make very low intensity light visible to a human being (or a computer system).

In this process, the wavelength of the incoming light is lost and only the intensity can be captured. This intensity appears green in analog systems because it's the wavelength at which the phosphorus plate (in green in the image above) emits light when hit by the accelerated photons.
Here is a light-enhanced image of a cute animal, what is displayed is the light (especially the one in the near-infrared region of the spectrum) reflected by it.

2. Thermal vision
Thermal vision devices capture light in a much lower wavelength than the visible light spectrum, it captures infrared radiation. It renders it sometimes in pseudo-colors for ease of use but the light it captures has no "color" to start with.
Here is a thermal image of a cute animal, what is displayed is the intensity of the "heat" generated or reflected by it.

A: The colors we see in a day-to-day basis are the result of a source emitting light in a broad spectrum with varying intensities for different wavelengths. If you shrink that spectrum to specific wavelength or to a narrow band of wavelength, then your experience will be of seeing a specific color in different tones.
Night vision cameras achieve their results by shifting a narrow band of infrared light into the visible light spectrum, but because it is a narrow band your experience will be of seeing a color.
A: 
Why doesn't night vision have color?

The normal human eye has two basic kinds of photoreceptors, rods and cones. Amongst the cones, there are three subtypes, each of which is sensitive to a different range of frequencies of visible light. These three different types of cones are what lets you see colors, but only if the lighting is sufficiently intense. Our cones don't work well in low lighting conditions. They shut down when the lighting is too low.
The rods on the other hand are much more sensitive to light than are the cones. The rods function even in extremely low lighting conditions. However, there is only one kind of rod. All that the rods can indicate is the presence or absence of light.

I misread the question as being about night vision in the human eye as opposed to night vision video cameras. A typical digital color video camera works by having an array of red, green, and blue color filters atop the array of image sensors. Even with those filters in place, the video camera oftentimes has to stop down the iris to avoid overexposure. Those filters merely hinder light collection when lighting is very low and the iris is fully open. This alone makes a monochrome camera a better choice than a color camera for night vision.
The image sensors' photodetectors are sensitive to ultraviolet, visible, and near infrared light. Ultraviolet and near infrared light would inevitably lead in color pictures that don't reflect colors as seen by the human eye. To avoid this, lenses and/or filters in a typical color camera block almost all ultraviolet and near infrared. There is no need for that in a monochrome camera. In fact, having lenses that transparent in the ultraviolet and near infrared enables those night vision cameras to collect significantly more light.
