Think about the light you need, either as a creative solution, or because it’s too darn dark!

Chances are you have either a built-in flash on your camera, or hopefully, a separate unit.
Now, we’re not going to get into studio lighting as that is a massive subject, and just want to look at the more portable illumination here.

As a start, for events you really should use a separate flash, and you can either go to the camera manufacturers proprietary units, OR go high end with Profoto, OR more affordable Godox.
Let’s first look at the on-camera flash, units that are mounted onto the camera, as above.
One thing that I’ve always preferred, especially in daytime events, is a flash directly on camera with exposure compensating for the background and allowing it to still be seen and not throw your subject in a black hole.
Like the photo above, and below:

You can either mount the flash right into the hot shoe, as you can see above,

or use a custom bracket placing it on the same level as the camera,

or use a transmitter/ receiver situation, or  direct cable from the flash to the camera body, and using the flash off to the side, above, bounced or however you have to adjust for the environment and your personal style.

Here is how Vivian Maier did it:

And a photog from the recent Oscars:

Let’s go all Goldilocks on your possible options.

The Small:
I wanted to bring you a compact solution using a Fuji X100 series camera, and what I think, is one of the best tiny flash units out there: The EF-X20. Here it is mounted on the top. As you can see by the dials, you have a TTL option, and the lever on the side widens the flash reach. Don’t hate me, but it may only be available via EBAY. Worth it!

Have a look at this compact set-up below, and as you can see, you can either mount the small flash in the hot show, or for an off-camera flash lighting, adding in a set of Light Q transmitter receiver. So tiny. One thing to note is that they only work with the flash in Manual and not TTL. (the Light Q here is V1 and improvements may have been made in subsequent versions)

If you know how you shoot, and can assume your exposure consistency, you are good to go.
Yes, of course, determine your preferred distance from subjects, and desired DOF.
Flash runs on 2 AAA batts, so bring plenty. Recycle time is not stellar, but not the point with this set-up. Lower the output, and move closer.
The Medium:

Throw on a Fuji EF-60, or a Godox. Solid output, and you can just go with a transmitter on the camera if off-cam shooting, or just pop it on the body.
Wireless transmitters that does allows for TTL are the FUJIFILM EF-W1 Wireless Commander, that lives on the body, and can fire off the flash from the camera without a second receiver on the flash.

Batteries again, and of course use the rechargeble eneloops or similar. Bring a bunch!

The Large:
In this scenario, the camera is full size, and a Profoto A10.

Profoto makes transmitters specifically for your camera, so make sure you get the right one.
With a fast recycle time, rechargeable batteries, beautiful menu, and a round head, this is a main choice for an event where you will be able to probably shoot a 3-4 hour event with 4 rechargeable batteries. You can see above that I put a label on a rechargeable battery as there are 2 capacity options, A1 and A1X.

Also, with a flash of this size, you can add bounce cards/reflectors to the unit to soften or simply customize the light you want to work with.
As with all camera gear, what you use, depends on what you’re shooting. PLUS they make a wireless transmitter to sit on top of your camera. Connects immediately! Once again, match the transmitter to the flash and you camera brand.

In my EDC, is a Fuji x100, the EF-X20, and the Light Q transmitter/receiver set. The  option.

For a family/ friend’s event, you may want to go medium, so the gear is not overwhelming for your subjects. C’mon, you know them and want to keep it light (sorry).
For a full-on event- indoor or outdoor, wedding, business event, music (probably rock only, and be judicious with your flash!) even a portrait in a pinch.
This becomes, without question, the large option. For me the quality of the light, recycling time, and dependability,
Yes, it’s all about the look you are hoping for.

Now , it also comes down to how you are mounting the flash, if on camera.

There are 2 brackets I swear by- The Custom Brackets CB Mimi-RC.

Small, single pice brackets that holds the flash just in front of the camera body.

The other secret weapon I carry is the Newton Bracket. Not made anymore, but so well machined, and allows you to shift your flash from top to side, depending on your camera orientation. I usually don’t like sharing gear that is not currently made, but they have some on EBAY. Another item worth the hunt in my opinion.

There you have a basic breakdown of gear I use, and different combos.

There are other light modifiers I carry, but this collection are the basics, and used situation dependent.

Always remember to test your set-up before shooting, and bring plenty of batteries!