The Muay Thai roundhouse kick is probably the most common and powerful kick that exists. To throw your traditional Muay Thai roundhouse kick, you want to pivot your standing foot, followed by rotating your hips, leading with your knee, driving your shin horizontally to diagonally across your target and snapping the kicking leg. The goal is to kick through your target. It also helps to balance your body when kicking by whipping your arm down to counterbalance.
One method on how to throw your roundhouse kick is the gallop roundhouse kick - which is probably one of the most powerful kicks. To perform this, you will lift your rear leg up in the air, then back down as you step forward with your lead leg and into your kick. It's just a small lift up in the air and back down to develop momentum and power into your kick. You can also lift your rear leg up and forward before taking another step with your lead leg to close the distance on your opponent.
As you can guess, this style of kicking can be used to close the distance and generate power. The disadvantage of this kick is that it is telegraphed, meaning your opponent may read a roundhouse kick coming from that initial lift of your knee. At the same time, you can also use that initial lift of the knee to fake a roundhouse kick and attack with another weapon!
Another method to throw your roundhouse kick is called the shuffle roundhouse kick. To perform this one you're going to want to push off your rear leg and step 45 degrees to the side with your lead leg. You will then replace your lead foot with your rear foot, as you step to the side with your lead foot again and drag your rear leg back to load up your roundhouse kick. This allows you to kick your opponent at an angle, making it slightly more difficult to defend. Your opponent will also be facing the wrong direction at this point, so they should be stepping and turning clockwise to square back up with you. As they turn towards you, they are actually walking into your kick, causing more damage. The only downfall to this style of kicking is that it may require more energy to execute.
Thirdly, a really popular and unique one is called the question mark kick. This kick was coined by Muay Thai legend, Saenchai, and uses the question mark kick to land on the head. To throw this kick, you want to lift your rear knee up as if you were about to throw a push kick, but instead pivot your standing leg and whip your kicking leg into a roundhouse kick. You should be leading with your heel and turning your shin completely horizontal. This style of kicking definitely has less power to it, but it's much faster and sneakier because you are faking a push kick before turning it into a roundhouse kick. Because this kick is more of a snap kick rather than kicking through the target, it has a better retraction to it, meaning if your opponent catches your kick, you are able to pull your leg out of the catch and quickly get back into your stance.