What triggers are needed to fire a YN568ex III for canon 80D?

You have a few choices. Optical or radio? TTL & HSS or manual only? Built-in or add-on?

Radio

Assuming you want to use wireless radio control, rather than wireless optical control (and you do), your easiest choice is probably the YN622 system. You could also use the manual only Yongnuo YN560/RF605/RF603 system, but you'd give up some of the advanced capabilities of the YN568EX III such as TTL. Just to make matters more complicated, The YN622 and YN560 systems are partially, but not fully cross-compatible. YN622 items made since the beginning of 2015 can be set to receive signals from YN560 transmitters, but YN622 items can not transmit YN560 signals. YN560 receivers can not receive YN622 signals.

If you choose the Yongnuo YN622 system of triggers, you need the Canon versions of the YN622 series of transceivers and transmitters, such as the YN622C II or YN-622C-TX. Anything with an "N" instead of "C" in the model name is for Nikon cameras.

The YN622C II is a transceiver. That is, it has both a 2.4GHz radio transmitter and a 2.4GHz radio receiver. You can use one on your camera as a transmitter and another attached to your flash as the receiver. The transceivers also have a hot shoe on top as well as a hot foot underneath, so you can put the transmitter's hot foot into the camera's hot shoe and then attach another flash on top of the transceiver if you also want an on-camera flash. (I tend to use a flash bracket with on-camera flash, so I can use a transmitter without a hot shoe and control the flash attached to the bracket via radio, though sometimes an off-camera hot shoe cord works better for flashes on brackets attached to the camera.)

The YN622C-TX is a 2.4GHz radio transmitter only. It can transmit just like a YN622C II transceiver, but can't receive anything via radio. It also has a screen and more buttons that make it easier to change settings and see what settings you have selected. You can use one YN622C-TX on your camera and one YN622C II attached to your YN568EX III flash as a receiver.

If you've only got the single Yongnuo YN568EX III at this point, you might also consider getting a Godox 2.4Ghz transmitter (there are a number of different models available) as well as a Godox X1RC receiver to attach to your flash. You'll only need a Godox receiver for non-Godox flashes, like your YN568EX III or any other Canon compatible flashes. Most Godox flashes have built-in receivers for the Godox 2.4GHz radio system. Many Godox hot shoe mounted flashes also have built-in radio transmitters that can control other Godox 2.4GHz devices. So when using a Godox flash or transmitter on the camera, Godox flashes will have built-in radio receivers inside the flash.

The reasons I would recommend using Godox instead of Yongnuo:

• Godox has the same radio system and protocol for all of their 2.4Ghz flashes, all the way from their smallest manual only speedlights to their largest battery-powered monolights, and everything else in between. Yongnuo has two different major protocol systems (YN560/RF605/RF603 for manual flashes and YN622 for E-TTL flashes) plus they make another line that clones the Canon RT system.

• In my experience, Yongnuo radio triggers are very reliable within their limitations (YN622 transmitters can't control YN560/RF605/RF603 flashes/receivers, etc.), but their speedlights are not. If you use several Yongnuo flashes for any kind of paid work, you better have a spare or two with you, because eventually you'll need it. If you do anything that is remotely thermally challenging, like using HSS for more than an occasional pop here and there, you'll eventually have one crap out on you at the most inopportune time.

• In my experience Godox and Flashpoint (Adorama's in-house version of Godox products) flashes are more reliable than Yongnuo flashes, even though their cost is competitive with Yongnuo when one considers the internal Li-Ion batteries supplied with many of Godex's flashes are as good as using an external battery pack with AAA battery powered flashes.

• If you go with Godox you'll not have to worry about changing radio systems if you want to move to more powerful lights. You will also be able to use your speedlights and other Godox lights together at the same time using only one transmitter. Godox (and other, rebranded versions of Godox products, which all work together even if the different parts have different branding on them) is the only manufacturer at this time that has a single radio system for small portable speedlights all the way up to powerful mains powered studio flashes and powerful Li-Ion battery powered portable monolights.

In the United States, Godox devices are rebranded for Adorama's in-house Flashpoint nameplate. In the UK and Canada, Pixpro and StrobePro are rebranded Godox nameplates. Some (but certainly not all) Neewer flash products are rebranded Godox products. At one time Cheetah sold some rebranded Godox products, but started moving away from that in 2017. There are others in Europe who sell Godox products under various nameplates as well.

Optical

If you're willing to put up with the disadvantages of optical wireless communication, you can simply use the pop-up flash of your 80D to control an off camera EX flash, such as the YN568EX III. You'll probably soon discover that the limitations are too confining, though.

Some of the disadvantages with using optical wireless communications instead of radio communications:

• Distance limitations. Most optical systems, especially when using a weak built-in popup flash as the controller, are much more range limited than radio systems.
• Positioning limitations. Most of the optical controllers only cover an area about as wide as a 24mm lens on a FF camera. If the remote flash is further to the right or left it may not receive any optical signal, even if it is only a few feet from the camera! Radios transmit in all directions from the camera.
• Line-of-sight requirements. In addition to being in the "cone" of light transmitted by the master, off camera flashes must have a clear line-of-sight to the master with the optical receiver on the flash pointed in the direction of the master. This inhibits being able to place optically controlled flashes inside modifiers, placing them behind objects in the scene, etc. Radio systems are not limited to line-of-sight and can even be used on the other side of walls and other obstructions (although the obstructions may reduce the range somewhat).
• Difficulty with bright ambient light. Especially outside under sunlight, the power of optical wireless control is very limited. Again, especially with a relatively weak built-in popup flash, the master just doesn't have much power to cut through the bright sunlight and the receivers can't detect the weak signal from the master over the very bright sunlight. Radios work just as well in bright sunlight as they do in a dark studio.

About the only advantage optical control using the 80D's built-in popup flash has over radio control for you is that you don't need an on-camera radio transmitter attached to your camera's hot shoe. (But you'll probably soon discover that you'll need a more powerful on-camera optical master attached to your camera's hot shoe to get the optical system to work the way you want, if it will even work then.)

For these reasons, it is probably better to start by just choosing a radio system and then stick with that system. Optical control of off camera flash is fast becoming old technology that is only included in current products to allow them to work with cameras and other flashes that were made before the use of radio communication became more prevalent to control off camera flashes.