There are two different types of flashes: speedlights and studio flashes. For video recording they have there own continuous light, which I am not going to speak about.
There are different types of studio flashes: AC powered, battery powered and generator (some lights you can charge and shoot at the same time).
I think the most important thing is flash sync speed. It means, the time that the shutter will open and close and the flash will exposure the image. If you can see only half of the image you need to choose a longer shutter time. Usually it is around 1/200, but not always!
Second important thing is that flashes produce white light, were the white balance is similar to daylight, so if you use different light sources you will see it in the image. Speedlights usually come with tungsten and fluorescent filters that you can put in front of the light to even out the colour difference.
You have different options with speedlights like in cameras. You can have automatic, manual or TTF. TTF means that it will send pre-flashes to measure the light that is needed.
You can remove the speedlight from the camera and use it in slave mode or with a transmitter. Nikon cameras have the possibility to use the cameras own flash as a transmitter so you are able to use the speedlight as a studio light. I have used this a lot before.
Studio flashes need a transmitter, or an old fashioned sync cable. I recommend the transmitter. Usually they have it with the studio set.
Studio lights have a control light. With the control light you can see where you will put the flash. I don’t shoot with the control lights because they are usually tungsten a yellow light.
There are different types of light shapers in the lighting (I could write own blog post about this)
Umbrellas: White, silver or mixed with gold. Shooting through, reflecting or using both. It is light weight and easy to use.
Reflector: There usually come with studio flashes. There are different depths and a bit different models.
Barn doors: Simply black metal pieces on the lights and it will control the light width and high a bit more. You can have these on soft box, they are usually fabric and used in front of the light.
Softbox: There are different shapes and sizes and depth depending the brand. You can have these in small places and depending the brand it can be easy or hard to pile. It can produce soft light. Smaller soft boxes can be used in smaller places.
Beauty dish: It is hard light but beautiful. Today there are foldable versions. There are metal or white ones and different sizes. In front of the light you will have a spoon and it will scatter the light. You can have softer or grids in front off it.
Snoot: Cone shaped that will target the light on something. It is quite handy if you have a product photo session.
Grid: Not really a light shaper by itself. A simple grid or a comb in front of the light, you can have it in beauty dish, soft box, reflector or snoot. There are different sizes and densities.
There are smaller versions of these for speedlights and different sizes for studio lights.
Online you can find how you can built these at home, incase you want to test some of these. You don’t what to put anything directly on the light. Remember that the flashes can heat up quite a bit so it can melt or burn.