There are two basic types of strobe flash that the beginner can purchase rather inexpensively and use. One is the speedlight style flash. These are always battery powered, small, and very portable. They typically are the less expensive option too. They have two parts, the actual strobe head (which usually can tilt and swivel) and the control and power portion below. There will be a hot shoe on the bottom that will allow the flash to be attached directly to the top of the camera. You can also use the speedlight off camera. Not all speedlights are the same. We will touch on some of the differences here, but it will not be an exhaustive list. The speedlight is very effective, but they are the lower power option in strobe lighting. The speedlights will fall into two basic groups for max power levels. There are smaller lower power units that are pretty close to each other, and then upper end power that are all pretty close to each other. Obviously the more powerful lights will be a bit more expensive. Prices range from well under a hundred dollars to around five hundred dollars top end.
The other main style of strobe lighting is the monolight. These are typically AC powered instead of battery powered, however you can get battery packs with an AC inverter to be able to use them where there is no AC power. The flash units are more expensive than the speedlight style. They also are more powerful too though. A monolight can go from a few hundred dollars to well over a thousand dollars each for lights. The biggest benefit with a monolight is the AC power and not needing to worry about batteries. But they have some other huge advantages though. They typically have a faster recycle time between flashes. This means from the time the flash fires till the time the strobe is recharged and ready to trigger again is much shorter. They also have a modeling light. The modeling light lets you see what the light is going to look like on the subject before you take a picture. It is a low-ish wattage bulb, usually between 60 and 100 watts. So it is easier to visualize the look you will get.
Another big difference between monolights and speedlights is that there are almost no monolights that do TTL. So you are limited to manual power setting only with the monolights. There are a couple of exceptions, but as a rule this is the case. It is not overly difficult to learn to shoot manual, and many photographers prefer to shoot manual lights instead of TTL. But it is something to consider.
By far the best option is to use a wireless trigger. What you have with this is you have a radio transmitter, or transceiver, that goes into the hot shoe of the camera. Then you have radio receiver on one or more of the flashes. Again, you can optically trigger some if you want. Then when the camera shutter is pressed the wireless transmitter will send a signal to the receivers and tell the flashes to go off. The wireless triggers are much more reliable than optically triggering, and will work for great distances. You also don't have to have the line of sight between flash units you need with optically triggering the flashes.
If you are using manual flash units, either speedlights or monolights, then you can get a very basic wireless trigger set up. These are very inexpensive. Typically to connect to a monolight you use a PC cable from the trigger to the sync port on the light. With the speedlight the wireless receiver will have a hot shoe that the speedlight can slide into.
There is a lot more to talk about with off camera flash. But this will get you started with understanding the basic equipment. It seems scary, and it is a huge step forward in your photography. One suggestion is to start with a single off camera flash. Don't try to get crazy and start with a multiple light set up. Learn and master a single light. Then add more to the mix. You can do amazing things with just one light.