A Twist Kick (also called a Twisting Kick or biteureo chagi) is a bit like a Roundhouse Kick that's performed inside-to-outside rather than outside-to-inside. Like a Roundhouse, the Twist Kick can be performed with either the ball of the foot or the top of the foot.
To perform a Twist Kick, you raise your knee just as you would for a Roundhouse Kick or a Front Kick, but instead of turning your hip inward as you would for a Roundhouse Kick, you turn your hip outward instead, and kick outward away from the body. To make this possible, the off-leg must often be bent into a slight crouch, with the off-foot turned in the opposite direction of the kick, in order to achieve the necessary hip rotation.
Example: Assume the right-legged Kicking Stance, i.e., with the right foot back and the left foot forward. Raise the right knee as if to perform a Roundhouse Kick. Just as with a Roundhouse Kick, the left foot will pivot so that the toes point to your left. The right foot, however, instead of moving from right-to-left to cross in front of your body moves from left-to-right away from your body, out toward your target.
As you can see from this example, in a Twist Kick the opponent will not be standing in front of you as he would be with a Roundhouse Kick or Front Kick. Instead, the Twist Kick is a way to kick an opponent who's off to your side. The same could be said of an Outward Crescent Kick, but the Outward Crescent Kick requires much less hip rotation since it uses the edge of the foot (the "blade") as the striking surface.
The Twist Kick can also be performed as a Jumping Twist Kick in order to achieve more height.