Most lighting equipment falls into one of three different classes: Light Sources, modifiers and grips
Grips expand to encompass all the stuff we use to support and hold our gear. Be it your camera, or lighting equipment such as stands and clamps.
Light modifiers in studio photography are anything we attach to (or hold in front of) our flash or light source, to control the light pattern it normally emits. These would include umbrellas, softboxes, and reflectors.
Light sources are any number of things that emit light onto our scene. It could be available light such as the sun or tungsten light, or it could be artificial such as flash units and strobes .(Source: http://photography.tutsplus.com/articles/an-introduction-to-indoor-and-studio-flash-photography--photo-4229)
Light modifier: What is a softboxe?
A Softbox is a type of light modifier, one of a number of photographic soft light devices. All the various soft light types create even and diffused light[ by directing light through some diffusing material, or by "bouncing" light off a second surface to diffuse the light. The best known form of bouncing source is the umbrella light where the light from the bulb is bounced off the inside of a metallized umbrella to create a soft indirect light.
A "softbox" is an enclosure around a bulb comprising reflective side and back walls and a diffusing material at the front of the light.
The sides and back of the box are lined with a bright surface - an aluminized fabric surface or an aluminium foil, to act as an efficient reflector. In some commercially available models the diffuser is removable to allow the light to be used alone as a floodlight or with an umbrella reflector.
A softbox can be used with either flash or continuous light sources such as fluorescent lamps or "hot lights" such as quartz halogen bulbs or tungsten bulbs. If soft box lights are used with "hot" light sources, the user must be sure the soft box is heat rated for the wattage of the light to which it is attached in order to avoid fire hazard. (Source: http://en.wikipedia.org/wiki/Soft_box)
Softboxes come in various sizes for different lighting circumstances. The larger the light source is relative to the subject, the softer the light will be. In addition, the closer a softbox is to the subject, the softer the light will be. For this reason, small softboxes are typically used for small object photography or for dramatic lighting of larger objects. Larger softboxes are most often used for people photography. They can usually collapse for easy storage or transport. (Source: http://www.studiolighting.net/what-is-a-softbox/). The softbox accessories are grids and round and square mask, placed in front of the softbox. They provide a more directed light.
Square softboxes gives an edge control and maximum surface area coverage. They are suitable for many types of photography: portraits, still life photography. Square catch lights.
The octogonal softbox is recommended as key light. It will create round catch lights which look more like the light is coming from a natural light source. They are used by fashion and portrait photographers for their ability to provide very soft output in a relatively shallow profile design. They produce light that “falls off” very quickly as it gets further from the source, so backgrounds can remain less affected than if they were lit by an umbrella. It is used for single subject portraits and group photography.
The striplight is great for a hair light. It creates side lighting in portraits or full body beauty photography. They can be used as portrait light, rim light, hair light as well as sweep light (Used to top-light objects on a table or sweep). Mounted vertically, a strip light will illuminate a head or half-length of an individual. But, unlike a square softbox, it won't spill much light onto the background.
The rectangle shape softbox can be used with a main light source or a fill light. It creates square catch lights which look like the light was coming from a window. They are used in Product photography, Beauty photography.