Most of the noise in a digital picture is due to thermal noise. This is due to the random motion of electrons at the operating temperature of the camera, this is called Johnson–Nyquist noise and this is proportional to the square root of the absolute temperature. This means that changing the operating temperature from 20 C to -20 C will reduce the noise by slightly less than 8%. So, you have to cool the camera to extremely low temperatures to eliminate the thermal noise. In practice one uses liquid nitrogen cooled sensors for applications where extremely low noise levels are needed, see here for an example of such a camera sensor.
Another source of noise is due to the finite number of photons that will lead to some finite number of electrons in a photo-site. This leads to two sources of noise, the number of photons per unit time that hits a photo-site emitted from a constant power light source will fluctuate. If on average N photons will hit the photo-site during the exposure, then the fluctuation will be of the order of sqrt(N), this means that the relative fluctuation scales as 1/sqrt(N). So, the lower the intensity of light, the more important this source of noise is.
Another effect leading to noise is due to the stochastic nature of the interaction of photons with electrons. The same number of photons hitting a photo-site will not always lead to the same number of electrons being moved into the photo-site. The net effect of the finite number of photons releasing a finite number of electrons then leads to a net amount of noise, which is called shot noise.
Shot noise is the dominant noise source in pictures taken by professional liquid nitrogen cooled cameras like those of the Hubble space telescope, it doesn't play a role in ordinary digital photography where it is always swamped by thermal noise.