Side-to-Side:

This is a precision, action exercise that challenges your balance, speed, tendon strength and even your aerobic ability if you do it long enough.  

What's involved: Front hip flexor, lower back, glutes, side hip flexor, lateral abdominals, quads, adductors, calves, hamstrings, cardiovascular system, aerobic performance (VO2 Max) if you do it long enough.

Powered by: Quads (primarily) and lateral abdominals (to maintain the precision). 

Tip: Watch in the video the positing of the arms inside the body. One arm is parallel to the kicking leg, while the other one is tucked inside the body's circumference. This is key to maintaining perfect posture. 

Double Side Kicks / Low & Mid:

https://www.youtube.com/watch?v=zZz30fcHbvE&

Double side kicks at two different heights challenge your balance, tendon strength, core stability and flexibility. Do enough of them and you also find yourself gaining in endurance. Competition martial artists go through 1,000 rep sessions, where they perform 1,000 double-side kicks in 100-kick lots. If you're ever looking for a challenge set aside some time and give it a try. .

What's involved: Glutes, core, side hip flexors, adductors (for flexibility), calves, quads. 

Powered by: Core and glutes (primarily) and your quads (as a final flick). As an exercise this engages most of the major muscle groups in the mid and lower body.

You can seriously improve on this by paying attention to your posture. Lean towards the kick when you kick (do not allow your upper body to drop away) and extend the arm that is on the side of your kicking leg, so that it points to where you are kicking. Tuck the other arm inside the circumference of your body. This helps you maintain your balance.