Wow, this is an extremely broad question, as there are dozens of light modifiers, but I'll cover the most important ones for strobe lights.
Generally speaking, light modifiers are all about controlling light. Whatever you decide to do with, you want to keep a reign on it. As such, we can split modifiers into two main groups: soft and hard modifiers.
Modifiers that soften light generally work by diffusing it - spreading it out over a large surface so that light rays are scattered. They are used anywhere you want to reduce shadows and make things look smoother. Their number one use is for key lights in people photography.
The most popular of these modifiers are umbrellas and softboxes.
- Shoot through umbrellas - these are semi-transparent umbrellas that you place in front of your light source. They work just like putting a lampshade over a lamp's bare bulb, you place them between the subject and the strobe. Imagine that they "filter" the light like a sieve.
- Reflective umbrellas - these are opaque, black outside and are usually silver, gold or white inside. These are placed open-side to the subject, with the strobe pointed away from them, straight into the umbrella. The inside reflects the light, scattering it in every direction. The various liner materials determine the color and quality of the light being reflected.
Umbrellas come in all shapes and sizes, from tiny ones to ones over 3 meters across, and are easily the most popular modifier you will find. Their main advantages are their small size, low price and versatility - you can shoot with a folded reflective umbrella for a very specific light quality. Their biggest disadvantage is a lack of control they give over light spills.
There is really no home-brew solution for umbrellas. They are extremely cheap (< $10), which is about as much as a real umbrella you would have to hack up costs.
- Softboxes - these are usually pyramid-shaped constructions, lined with a silver material, with room for a strobe at the top of the pyramid and diffusion cloth at the base. They come in almost all dimensions, from tiny inflatable(!) ones for speedlights, to huge, studio-only setups that require multiple light stands to hold. Their biggest advantage is their ability to control light spill and to accept modifiers. They are heavier, bulkier (even folded down), more expensive and take longer to set up than umbrellas.
- Octas - there are two versions of the octa: either as a softbox with a octagonal shape, or as a hybrid softbox-umbrella, which is what I'll be writing about here. A strobe is placed inside something akin to a deep umbrella, then diffusing cloth is placed behind it. In practice, this creates extremely soft light as light is first scattered off the reflective material inside the "umbrella", then diffused again by the cloth. With large octas it's difficult to support the entire modifier off the front of the strobe, so some manufacturers (Elinchrom, Profoto, etc) have octas that mount on the light stand and allow you to place the strobe inside it.
Softboxes are what most people think of when they think of studio photography, so they make a good impression if someone's coming to pose for you. Using additional modifiers, like a grid, flag or filter on top of your softbox can give you even more control.
The key in softboxes is diffusion, you don't need to literally make a softbox, you can throw a bedsheet over a broomstick supported by two chairs and you'll make yourself an excellent, soft light source. You can literally make a soft box by lining a cardboard box with aluminum foil and pulling a sheet over it, but the comedic effect while shooting portraits might be a bit too much.
Hard light is the default with strobes - that's what we get when we fire off a speedlite or on-camera flash. A small point of light that brings out imperfections and coarse textures. Shoot it from the side and it'll make a child look like they have awful skin. Still, it's extremely useful to have controlled beams of light (think: rimlights/hairlights) and it's absolutely critical in product photography.
There are four basic types of hard modifiers:
- Reflectors - these are the standard bowl-shaped accessory that pretty much every studio strobe sold today ships with. Their only use is to cut down light spill to the sides when using a bare bulb (which is rarely done) and when using umbrellas or grids.
- Grids - these are literally metal grids of various thickness and density. The black tubes they form work extremely well to stop scattered light rays coming from the strobe bulb, creating a thin, relatively soft beam of light. The quality of light is determined by the density of the grid, while the size of the beam (expressed in degrees) is determined by the thickness of the grid. Grids often also include gates, which are movable flaps on each side of the grid which allows for finer, tighter beam control.
- Snoot - a very simple construction, this is usually just a slightly tapered tube of black metal. It creates a very harsh, small beam of light.
- Beauty dish - this is the most complex and - in my opinion - nicest of the hard modifiers. A small reflector (usually silver, gold or white) is placed in front of the strobe, while a bowl-shaped reflector is placed around the strobe. This creates a very even, hard light with an extremely sharp drop-off. Used almost exclusively in people photography, it's extremely useful as a key light, especially for people with good, smooth skin (or makeup). As the name implies, it is extremely popular in beauty photography.
As far as DIY solutions, Strobist has a huge DIY section, but be warned: a lot of the store-bought stuff is a lot sturdier and gives better results than the DIY equivalents. You can fake a snoot on a speedlight using a cardboard tube painted black on the inside, but past that... just get the chinese knock-off. Again, instead of recreating these modifiers in a home setting, look at what they do with the light and try to emulate that. A single soft light source as a key, a big piece of white paper as a reflector will give you very nice results. You can always cut up the paper, replace part of it with aluminium foil, put a flag between your model's eyes and the light source, etc. The possibilities are endless.
Remember, light is just light, most of these modifiers can be (and are!) mixed and matched: grids on snoots, diffusion material on beauty dishes, grids on softboxes (these are almost always cloth), silver backs for shoot through umbrellas, etc. There are also tons of different names for everything I've listed above, I just used the ones I'm most common with.
Three critical components of studio work that I didn't cover as they aren't strictly light modifiers, but are worth talking over: flags, scrimms and bounces.
- Flags - anything that's not a light modifier that blocks light. Professional flags tend to be black cloth or paper stretched on a frame that has a small handle protruding to make it easier to place on a stand. At home you can use pieces of black paper and just tape them where they need to go, or maybe black cloth if you need to stop a white wall from reflecting. Used mainly to control light spill and keep down highlights.
- Scrims - these are panels covered in diffusing cloth, placed where light needs softened locally. You know those outdoor TV scenes where it's a bright, sunny day, yet all the actors are lit by nice, soft light? That's a huge scrim over their heads. At home, just use a piece of white cloth, you're good to go.
- Bounces - also known as bounce cards. Literally, anything that will reflect enough light to be useful, but usually opaque white, silver or gold material stretched out on a frame, but you you'll often see mirrors used for this. Usually used in contra to a key light as a way of lightening (or opening up) shadows created by the light. The type of material determines the light's quality, the distance from the subject and light determines the bounced light's intensity.
That should cover it, I'll update if anything else comes to mind.