Gyaku-tsuki, the reverse punch, is analogous to a boxer’s cross punch. Reverse punch is the most commonly used karate technique, so practice it well -- you’ll be seeing a lot of it.
To throw a reverse punch, run your arm along the side of your body, until your elbow is one fist-width away from your chest (like the starting point for a front-foot punch.) Then turn the fist over with a snapping motion, just like in front-foot punch. Remember to incorporate reciprocal action when punching by pulling the non-punching opposite-side hand to your hip, or performing a Kūsankū movement by pulling your opposite-side hand across your centerline to the side of the face to swat an opponent’s counterattack aside.
[video of reverse punch from the front and from the side, blocking face with other hand]
Contrary to popular belief, punching power is generated by the wrist’s snapping turn-over, by tensing all of the arm muscles at once. Power is generated by switching from a state of relaxation, to complete tension, and back to complete relaxation as quickly as possible. This is similar to cracking a whip, or snapping a wet towel. All Goshin-Jutsu Karate techniques incorporate this “snap.” Again, all techniques should be a total-body spasm-like reaction, similar to a sneeze. Other martial arts use snap techniques as well -- Bruce Lee’s famous One-Inch Punch was also driven by this same concept of snap. Any technique without snap is just a push.
[video of a side view of a series of reverse punches, hitting a kicking shield. The first three are arm-only, with no turn over. The next three are 6” punches. The next three are full-power reverse punches. Make it dramatic.]