Color Depth Comparison
 

Click on the above image to view at full size.
 
The original image with 8-bits per RGB color was reduced to 6-bits and 4-bits.
   4-bits per color: visible banding
   6-bits per color: banding is very hard to see (try to find it!)
   8-bits per color: no banding
 
One often reads that serious photo editing needs a computer and monitor capable of 10-bit color. That is only marketing BS. A monitor with 8-bit color will never cause banding by itself. As the above example shows, 6-bit color is usually adequate.

Note: 8-bits per RGB color is often called "24-bit color". 8 bits x 3 colors = 24 bit color.
 
Editing a photo to increase brightness can effectively reduce the bit depth, so a greater bit depth in the original image can be helpful. If a photo is brightened by 2 F-stops, the bit depth is effectively reduced by 2, turning an 8-bit original into a 6-bit edited version. That example would not produce noticeable banding, but brightening by 3 F-stops could produce noticeable banding in a monotone area like sky. This is an issue that advocates of RAW processing and greater color depth can rightly assert. The problem here is that RAW images rarely have more than 8-bits of low-noise color, so brightening a too-dark photo produces visible speckles.
 
The photo editing issue described above has nothing to do with the color depth of a computer/monitor system. Here 8-bits are fully adequate.
 
A more valid and frequent issue is the color fidelity of a monitor.