Not sure but many of the same things go with the same name.
Needed:
flash with manual power adjustment
something to trigger the flash (this can be an internal flash-sensing slave, a sync cord, or radio trigger). Remotes are convenient, but cheaper ones are less reliable and still more expensive than a basic sync cord. Cords are cumbersome and a trip hazard, but as long as you're careful with the connectors, hardwired = dependable.
cold shoe (preferably plastic mount so you don't short out your flash), this will screw on to something and hold your flash.
spud (a metal spud that fits into a super clamp or swivel to hold the shoe to an attachment device.
Options:
swivel (aka, umbrella swivel, etc.) receives the spud in a female receptacle, screws it in and holds it. The opposite end also has a female receptacle that mounts onto a light stand. Looks like
this.. Note that this one includes a male threaded and female threaded spud for your convenience (the male threaded spud goes into your hot shoe. The Female can screw onto a male ended tripod place or make a nice fishing weight.) Note also that they may differ in appearance.
Clamp (super clamp, multi clamp, justice clamp. JK on the last one) These come in many types too. Some are ratcheting types, some looks like Bessey clamps (the bit metal spring loaded ones at the hardware store), some look like vice grips... it's endless. The most popular is the Bogen Super clamp, a very durable (but under extreme conditions has been known to shatter). Attaches to pipes, tables, etc securely. Looks like
this.. Clamps and stands.... well most people just own both. Use a clamp where you cant use a stand, use a stand when there's nothing to clamp to. Most clamps do not include a spud.
Softboxes and shoot throughs are different beasts. Brollies have a softer light that spreads everywhere. They're much cheaper, easier to mount (with the swivel) and double as rain protection. They deploy much faster and most cases are more compact packed up. They aren't mount-specific so you can used them with small flashes on swivels or large studio flashes. The quality of light in a softbox differs (not good or bad, just different). They are more directional, require more care in aiming, and in most cases take more time to construct, don't store as easily (need to pack the backing, both baffles, a speed ring, and support rods). For small hot shoe flashes, you need a model that supports them. A few manufacturers make brackets that allow you to mount your flash via a series of metal adapters to fit the softbox, this only adds to the amount you have to pack. However, I prefer softboxes because I like the control of the light it offers. Though the majority of the time I use brollies because they are so darn convenient.