The bit depth refers to the digitization of the analogue light signal by the analogue-to-digital (A/D) converter of the camera, in other words, the binary range of the possible greyscale values of the produced images. It is sometimes incorrectly called dynamic range (which is actually a function of full-well capacity and noise). With increasing bit depth, the same signal information is simply being divided into smaller increments upon digitization. Common bit depths are 8 to 16 bit, which equals 28 = 256 to 216 = 65536 grey levels. Consider a chip with 18,000 electrons full-well capacity: 8, 10, and 12 bit A/D converters count 70.3, 17.6, and 4.4 electrons per greyscale, respectively (this is the conversion factor), if the entire full-well capacity is used. Sometimes the binary range is given in decibels with one bit equalling 6 dB (8 bit = 48 dB, 10 bit = 60 dB, etc.). See also Bit Depth of Digital Images.
In this context it is interesting to know that the human scotopic vision (night vision) has a bit depth of not even 6 bit (= 64). It can distinguish in the order of 50 grey scales. Computer monitors are able to display 8 bit digital greyscale images. 16 bit images are reduced to 8 bit upon display.
Considerations: How much precision is needed?
"More bits, more better!"? "Every bit counts!"? Or rather "bit depth overkill"?
This is easiest discussed with an example: Imagine a camera with 18,000 electrons full-well capacity and a camera noise of 10 electrons. The dynamic range would be 1,800. Now, a 12 bit A/D converter offers 4096 grey levels, that is already one bit more than needed to cover the given dynamic range. And if considering the sensitivity of the human eye, even a 6 bit black-and-white image might look "pretty". Furthermore, if one takes in mind that fluorescence is a rather weak phenomenon and photodamage a significant problem, in most applications only a small fraction of the capacity of the camera is used anyway. Accordingly, precision gain by opting for the maximum bit depth is partly fictitious. When quantification of signals is important, however, 8 bit cameras are inferior to cameras with a higher bit depth, at least for certain applications.

The examples show the limitations of the human scotopic vision. The cloudy sky with the smooth brightness gradients clearly reveals the bit depth differences of the three images. However, at first glance the eye is unable to tell the difference when looking at the pyramid with its small details. 8 bit or 3 bit seem to give the same result (actually only the 6 darkest of the 8 possible gray levels are used for the pyramid in the 3 bit image). The differences only become visible upon magnification.