Let's start with the definition of Diffusion: The spreading of something more widely.
The main purpose of diffusion is to spread light out to cover a wider area, such as providing coverage for a wide angle lens. And depending on the environment (a small room with white walls and ceiling versus a cavernous studio) it will also affect the contrast of a scene. If the diffusion is the same size as the light source it is covering it does NOT soften the light. Quality of light (hard or soft) is determined by the size of the light as seen by the subject. That is the transfer edge of the shadow where things go from highlight to shadow--an abrupt edge means a hard light, a long transition means a soft light. We can take a 2x3-foot softbox and put it 2-feet from our portrait subject and have a nice soft light. It we pick up that light and move it back 15-feet the light becomes hard. The physical size of the softbox did not change. But its size in relation to the subject changed. In close the large light provides its own fill light and "wraps*" around the subject giving the nice gradual transition in the shadows. Backed up it can no longer do this and you get hard shadow edges. This is all determined by the size of the light.
So, how does diffusion play into this? If you put diffusion material over the light that is the same size as the light the light is obviously the same size. And therefore the shadow edge is the same. However, by spreading the light out it will bounce off of more surfaces around the subject and will provide ambient bounce, both of which will lower the contrast giving you brighter shadows. But those shadows are still the same quality. Don't confuse contrast with quality. The farther you place your diffusion from the light the more softening happens because the diffusion material becomes the light source while the light behind it remains the light origin.
Diffusion scatters light. Diffusion can come from passing light through a translucent material such as the diffuser in a 5-in-1 reflector kit, or a white shower curtain, or scrims, or rolls of diffusion material from your camera store. It can also come from bouncing the light off of a mat surface like a painted wall, a sheet of foam-core, art board, etc. In most of these situations the diffusion is larger than the light source and therefor will make the light softer. All too often I hear photographers talk about putting two, or three, or more layers of diffusion right on the light expecting it to soften the light that much more. But all they are really doing is lowering the output of the light (which you usually do not want if you are running your lights on battery power). For light to be soft it has to hit the subject from many different directions. Once you have added a diffuser to a light it won't matter how many layers you add, it will not get any softer, just dimmer.
While talking about diffusion, let me also introduce the opposite concept, narrowing the light. This is done via the use of a baffle, such as a honeycomb grid on a metal reflector or a soft grid on a softbox. Snoots and barn doors can also be used to restrict the light beam. I bring this up here because I also see photographers put diffusion in front of their gridded lights or in front of barn doors. Diffusion in front of a grid or barndoors becomes the source of light and negates the effect of the baffle as shown in the above diagram.
Back to diffusion...
First, let’s go to the standard 7-inch reflector that is common to most strobe systems. In the accompanying photos on the left (click in the image to see a larger version) I have used the 7-inch on its own, then with one layer of diffusion clipped on, and then with two layers of diffusion clipped on. There are a few things to notice. First is the large shadow on the background. Without diffusion there are actually multiple shadows with distinct edges. This comes from the combination of the flash tube itself, the reflector behind the flash tube, and the dish reflector around the flash tube each putting out a little bit different amount of light from their different sizes. When we add diffusion the light is homogenized or evened out to cleaning up any uneven beam patterns so the shadow on the edge is more blended. Then look at the shadow of the nose and the highlight on the tip of the nose. By adding the diffusion (one layer, two layers, or a dozen layers) we do not see any changes to these from one photo to the next. The edge of the shadow is still the same hardness and the highlight is still the same brightness in relation to the overall exposure. As noted above the density of the shadow (contrast) might be a little brighter from light bouncing around the room, but that doesn’t affect the edge quality of that shadow. If we want to lessen the brightness of the highlight on the nose one solution is to bring the light in closer, which will make the light bigger and cause it to fall off quicker (See! Everything in photography is interdependent and full of trade-offs). As a specular reflection the brightness will actually stay the same while getting larger, but the exposure on the rest of the face will also get brighter and we will power down the strobe to compensate for that which will then also bring down the brightness of the hot spot. This effect will be more noticeable with a larger light source like a softbox.
Now let’s look at combining a grid with diffusion. Three examples again this time over on the right. The first is with the same one we saw with 7-inch dish in the above examples. Then the 10-degree grid is added which limits the spread of the light for a more dramatic look and deeper shadows because there is less light bouncing around the rest of the room. This is the look I suspect that everyone is looking for with the grids. But then some people put a diffuser over grid. And look what happens when the diffusion material is placed over the grid—the diffusion material does its job (scattering the light into a wider pattern, take a look at the diagram above) completely obliterating the effect from the use of the grid. And that is at the cost of two or more stops of light (translating into using more battery power if you are using a portable battery powered system). If you do feel the need to diffuse the light when using a grid you need to place the diffusion between the flash tube and the grid, not in front of the grid. We will take a look at how that works in the next set of examples.
This set was made with the Deep Zoom 11-inch reflector kit from Interfit. The Deep Zoom kit comes with a set of three grids (10-, 20-, and 30-degree) and a diffusion sock. Six examples this time. First we see the Deep Zoom by itself providing a nice crisp look. Adding the diffusion sock lowers the contrast, but retains the same shadow edge quality. Next I put on the 10-degree grid providing a more dramatic look with falloff of light across the background and a bit more contrast. Then I tried the grid with the diffusion sock over the grid and POOF there goes the grid effect that you spent good money on to purchase the grids. The next thing to try was with the sock behind the grid, between the flash tube and the grid. We get the drama back, but with a bit less contrast. So, while we’re at it, what happens wit with the sock behind the grid and a sheet of diffusion material in front of the grid. Again we can see that by adding diffusion in front of the grid we are negating the effect of the grid.
In all six examples take a look at and compare the transition edge of the nose shadow. By now I hope that you don’t really need to look at it to know that it is going to be the same because the size of the light didn’t change between photographs.
So, to recap...
Diffusion scatters light in all directions and makes it cover a wider area. As the light is scattered and some is absorbed by the diffusion material you will lose some power.
Diffusion material can be specific products like those from Lee and Rosco. Or it can be a sheet of tracing paper or a bedsheet or a frosted shower curtain. Here I am assuming that the diffusion is dense enough that you won’t see a hot spot from a light shining through it. This is usually called a full-stop diffuser. Your diffusion material should be neutral in color, but often isn’t. That white shower curtain might contain brighteners that make it cause the light to be more blue, or it might have yellowed with age. Be prepared to have to do some color compensation when processing your raw files.
The opposite of a diffuser is a baffle (though I am baffled as to why some softbox manufacturers and photographers call their scrims and the inner diffuser on their softboxes baffles). Baffles restrict the flow of light. Common examples are snoots, barn doors, and grids. Hard grids are available for use with metal reflectors like the 7-inch, the Deep Zoom, and beauty dishes. They are rigid with a honeycomb pattern of openings and snap onto the front of the reflector and usually come in densities from 10-degrees to 40-degrees. The lower the number the narrower the light coming out of them. Soft fabric grids are available for most softboxes and octa boxes--most Interfit modifiers come with soft grids included. They are usually around 40-degrees and allow you to have a large directional soft light.
Don’t confuse contrast (the difference between the light and dark areas of your image) with quality of light (how quickly the shadows transition from dark to light). While thinking about that, also do not confuse brightness with harshness. Moving a light in closer makes it brighter, but also softer as it becomes bigger in relation to the subject (or as seen by the subject). The exception to this is when using hard grids. Because the light rays are restricted by the honeycomb only the center rays from the light going through the grid will reach the subject, making the light appear smaller as it is brought in closer.
Diffusion scatters light
Baffles restrict light
Don't confuse light contrast and shadow density with light quality
If you want a softer light you need a larger light source. Adding diffusion can make the light softer as long as the diffusion is larger than the light you are adding it to. In this example we see what we get with no diffusion, with diffusion right on the light, and with diffusion larger than and separated from the light. From the first to the second image we can see that the shadow edge remained the same, but the shadows and the background got lighter in tone due to light bouncing around the studio. In the third image we see that making the light larger made the shadow edge transitions wider (hence softer).
*light travels in straight lines, it doesn't wrap around objects. But a larger light source will provide light rays from more angles allowing the large light to provide its own fill light.