I recently did a write-up on this very subject on another board. Since I'm not allowed to link to it directly, I'll copy-paste what I can to here:
The Cause:
The camera has a sensor in it that is basically a grid of little tiny photoreceptors. Every photon of light that hits a photoreceptor is absorbed and increases the electric charge in that photoreceptor. After the exposure time is up, all of the charges in each photoreceptor are dumped into circuitry that measures how much charge was in each cell. The greater the charge, the brighter the pixel.
Note that these photoreceptors only measure brightness and not color, so to get color you either have to place an alternating grid of filters over the sensor, or you have three sensors (one each for red, green, and blue) and a prism that splits light out to all three.
Noise comes from two causes, but the primary cause is photons aren't the only thing that can increase the charge held in each photoreceptor cell. Heat can randomly increase the amount of charge by just a tiny little bit. In direct sunlight this isn't too much of a problem since there are a lot of photos hitting the sensor and they tend to overwhelm the noise.
Low-light is where the noise becomes apparent -- there are fewer photons hitting the sensor so the charge amounts are run through an amplifier (so the generated image will be of the proper brightness) which also amplifies the effect of the random charge noise introduced by heat. The greater the amplification of the charge signal from the sensor, the greater the noise. This is why the more sensitive ISO settings on a DSLR camera are grainy. It's the same effect.
The other thing that causes video noise is the photons themselves. Photons bounce around willy-nilly so the number of photons will vary slightly between photoreceptors and between frames for the same photoreceptor. This accounts for some of the low-level noise you see in a daylight video shot. Once again, in low light the amplification applied to the signal from the sensor will also amplify the slight photon variations into larger variations.
The same thing happens in video filters. If you have an image or video with a little grain and increase the brightness or contrast it will amplify all the little variations and result in more visible noise. The reverse is also true: reducing brightness or contrast will lessen the noise. If you can overexpose your video (without blowing out your highlights) and then darken the exposure in post, that will reduce some of the visible noise. Note that this only works if you increase your exposure by altering your aperature or shutter speed. If your video camera brightens the exposure by simply amplifying the sensor signal (this is how cheaper cameras do it) then you'll gain nothing since the overexposure will introduce more noise from the start.
As for why you see different amounts of noise in different colors -- this might be due to variations in the physics of the sensor itself, but the main reason is because of how the human eye works. The eye is more sensitive to brightness than color, and is more sensitive to reds and greens than blues. Highly-saturated colors are bright, so your eye will pick out noise variations more easily than in darker colors.
The Solution:
Aside from only shooting in bright light (eg. daylight or a fancy lighting setup), I've only found one program that successfully combats video noise acceptably:
NeatVideo. I've been a user of their still-image noise removal program,
NeatImage, for years but I just recently purchased their video filter and it works beautifully.
The best way to use this program is to get a grey card (or just film a blank, solid area) out of focus (so the only thing the camera sees is noise and not details) at the same camera and light settings your scene will be (or was) shot in. NeatVideo analyzes the noise and can then filter it out of the video with a pretty staggering degree of success. It also has an "advanced" mode that lets you adjust how much filtering to do at different noise levels and lets you separately control how much filtering is applied to the brightness and color channels.
You can remove noise from existing video without shooting a solid out-of-focus shot as described above so long as there is a solid area in at least a single frame of video that you can select to tell NeatVideo "this is supposed to be a solid color, so analyze this part of the frame and treat all variations as 'noise'". This calibration can then be applied to the entire video clip.
It's an easy tool to use and if you only need to work with standard-def video, the Home version only costs $99. (I work with higher-resolution footage and so sprung for the $199 Pro version. Worth every penny.)
Examples:
(click the image for the non-cropped full HD frame)
Original:
NeatVideo noise reduction:
