You'll hear that a lot ... whether involving what sharpening method and how much or when, bit depth, curves or levels, ... it ALL depends. On what you are starting with, and where you want to go with it.
As to bit depth, 8-bit has only the infamous 255 levels or steps (plus 0), for a total available "information" depth (or height ... think of it as a vertical scale is easier for some people) of 256.
16-bit has a bit more detail levels ... my memory is faulty but there are PLENTY around here to correct my mistakes, but I believe it is something like 4,096 levels, in the same absolute dark-> light progression.
Jpegs and most printing devices currently take 8-bit as their norm, and it is entirely adequate for printing a full-range print for human eyes. In fact, if you have a 16-bit image, you normally need to take it down to 8-bit to print, though some newer printing devices can handle 16-bit files (though one article I read said that at least a couple of the models that can, simply down-mode to 8-bit in their own processor). However, an image with 4,000+ levels can be modified quite a bit and then fitted "down" into a 256-level format with no visible problems for the processing steps.
So when is 16-bit advisable? If you are going to need to do much adjustment of the image light/dark properties, say you are going to raise the mid-darks up to a mid-tone, or try to take highlights down to mid-highs and do that with an 8-bit, you'll have many of those steps that, because of moving or lifting the detail that once was there higher up or lower down the scale, now have NO information ... coning, it is called.
You see it in the histogram, where there is a narrow canyon in the middle of your block of "data" that goes clear to the baseline, empty. If you have a few of those in your print, especially in the near-highlights, you will DEFINITELY see them in the print, as the tones will jump with a sharp band from one value of lightness to the next, rather than transition smoothly from light to dark.
Also, if you do a lot of image retouching, you can get coning.
So, if you start your pp (post-processing) by switching the image depth mode to 16-bit, you give yourself a bit more "headroom", though not nearly as much as if you start out with a 16-bit file to begin with. Which would be a raw file from the camera.
Two additional things ... for basic snapshots, if you've got your camera down so that your exposures are pretty good, save time and space and shoot jpeg. And a bit of extra accuracy 'cause I KNOW I'll get corrected by someone around here ... most camera's actually shoot 12 or 14 bit raw files, but in software, it's always called 16-bit.
Hope I've muddied things up wonderfully well there!
Neil