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CCDs are getting pretty good these days, but all systems are subject to noise. A typical value for a CCD seems to be 20 electrons RMS per pixel. This article from qsimaging.com says

CCD Read Noise is a fundamental trait of CCDs, from the one in an inexpensive webcam to the CCDs in the Hubble Space Telescope. CCD read noise ultimately limits a camera’s signal to noise ratio, but as long as this noise exhibits a Gaussian, or normal distribution, its influence on your images can be reduced by combining frames and standard image processing techniques.

It doesn't really describe the source of the noise however. I know that astronomers like to cool their CCDs, so I'm thinking it is from thermal phonons being converted to electrons? I'm specifically asking about the CCD so lets assume that the shot noise of the incoming light is well below the typical CCD noise quoted above.

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Confusingly enough, what you desire is for the photon shot noise to be greater than the electronic noise, because then your image's quality is purely signal-limited (and there's no need to spend more money on higher-quality hardware, to put it bluntly). –  Carl Witthoft Feb 7 at 13:05
    
@CarlWitthoft Indeed. That is why even cell phone cameras take excellent pictures in bright lighting. –  Chris Mueller Feb 7 at 17:35

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There are a couple of main sources of intrinsic error (that is, not associated with counting photons from your source) which CCD's have. The first is as you have already mentioned called read noise. Here is a reasonable definition of read noise (taken from Romanishin's free pdf on Photometry):

After an integration (exposure), the CCD must be read out to find the signal value at each pixel - because the signal may be as low as a few electrons per pixel, this step involves some very sophisticated amplifiers that are part of the CCD itself (on chip amps). Unfortunately, but inevitably, the read out process itself generates some electronic noise. The average noise per pixel is called the read noise. Modern CCDs typically have a read noise of 5 to 20 electrons per pixel per read out (read noise is the same whether exposure is 0.1 sec or 3 hours).

The second is called thermal noise (again as you have already mentioned), and it is the signal which one needs to deal with by taking darks. Again from Romanishin:

If we allow the CCD to integrate for some amount of time, WITHOUT any light falling on it, there will be a signal (and more importantly noise associated with that signal) caused by thermal excitation of electrons in the CCD. This is called the dark signal. The dark signal is very sensitive to temperature (lower temperature = lower dark signal), and that is why CCDs used in astronomy are cooled (often to liquid nitrogen temperature). Even with cooling, some CCDs have a non negligible dark current. This must be measured and subtracted from the image. As for the bias, we want to take many dark frames and combine them to beat down the noise. (The dark frame and bias frames are NOT the same thing!)

Hope this helps clear up any confusion.

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Great answer. I would add a side note concerning "Fixed-Pattern Noise." This is a non-temporal effect representing the variation in the apparent signal (e.g. voltage level) across the pixels at zero input and zero integration time, i.e. zero dark current. –  Carl Witthoft Feb 7 at 13:06

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