8 bit refers to any number in binary from 0 to 255, in binary written as 0000 0000 to 1111 1111 (those are 8 “bits” or binary digits). 16 bit refers to numbers from 0 to 65535 .
When talking about sound, a sound is saved, generated, or sampled, using numbers ranging from 0 to 255 in 8 bit. This leaves a lot of soft sounds just being noise or silence. CDs use 16 bit numbers to encode music. It's pretty good.
In the image below I have made a picture of a waveform, one in low resolution (like 8 bit) and the other in high resolution.
The numbers represent how far the speaker is into or out of the cabinet during the sound. It is pretty clear that the higher resolution sound more accurately approximates a sound wave. Once you hear the difference you can’t un hear it.
In photography, 16 bit grayscale images can have more subtlety in what values can be encoded.
consider an image like this one with 8 bits per pixel.
The gradient strip started at 256 pixels wide. There should be one pixel for every value of gray. If we used 4 bit color resolution with gray, we would have the strip below with 16 levels of gray.
It looks bizarre because I disabled the “dithering” that scatters pixels of adjacent colors to simulate more smooth transitions. With the dithering, It looks like this:
There are many different values of gray. If we turn it to a 1bit image (only black or white) Photoshop, and other software can turn it into a spray of black and white pixels called dithering but the subtlety is gone.
Quora does some funny things with bitmaps so here is a detail
Now imagine an image with double the number of grays of the first egg image. The first image has 256 levels of gray. A 16 bit image 65,535 levels of gray possible.
I can transform the 8 bit egg image to 16 bit but there isn’t much point. It only makes a difference if I can capture 16 bits of data. Almost none of the tools that I rely on to modify images in Photoshop work in 16 bit mode.
It really might only be visible to professionals in a very well crafted print. To make good use of 16 bit images you need a lot of knowledge, and a good process of experimentation to see whether the changes you are making are changing your final image for the better.
You can start with a raw image and convert it to 16 bit in Photoshop. If you start with a JPEG you are going to add file size but not subtlety. Think of it this way; you start with an image of 4 bits, you only have 8 values, 0 to 7, if you convert a gradation from white to black to 8 bits, you still only show 8 values because the 8bit conversion only changes the representation of the values, not the source data. If you looked at graph of your converted image, where the x-axis shows the value and the y-axis shows the number of pixels, you would have eight spikes with nothing in between. The first spike would be at 0, then nothing until 31, then nothing until 63 and so forth, all the way to 255. This is called upsampling and decreasing the resolution is called downsampling. There are many ways to decrease the information loss with upsampling and downsampling, but there is no way to create (meaningful) data that was not there originally.
In the early days of video games, there was so little power available in the graphics hardware that they had to use tricks to get things to move and be colored and respond to user interaction. This is a great video on the subject
One of the tricks is to use a lookup table which stores a few mixtures of colors so you can have very specific colors, just not lots of them. Photoshop can do this using the “indexed color” mode. This is only 16 colors but they are chosen very carefully.
An image with 8 bit resolution for each of the primary colors has a possible 16 million combinations and we cannot see any more color resolution.
This image is from the web. The quality is questionable. The color resolution at each step limits the color of the output.
Monitors and phone screens have limited color resolution as well.