Throwing a punch is simple in theory, yet something that takes a lot longer then most people would expect to “get right”, let alone master.
One of the concepts that tends to go most against peoples instincts is power generation. The power has to come from the legs first, and arms last.
Most people tend to start off wanting to throw a punch from the shoulder, even leaning forward into it, but doing so will never generate a great deal of power. The punch ends up turning over, then going out, with the elbow flaring out to the side. It looks kinda like a one arm bench press with the person leaning forward.
What you need to generate force on a punch is similar to what you need to generate force on a push. If you are trying to move something heavy, such as pushing a car, what do you do? You bend your knees, get your elbows in front of you, dig your toes in and drive from the legs. If you tried to push with your elbows out it doesn’t matter what your legs and core do, you can’t get more force then what your arms alone can generate do to the angle. If you bring your elbows in arm strength is no longer the weakest link in the chain as your skeletal structure can support more weight and let your legs and core drive first and add the arms on top of that.
The power on the punch, or push, comes from the back leg. In the same way you wouldn’t generate much force standing on only your lead foot and pushing something, you can’t generate a great deal of force doing the same thing while punching, the power behind the motion needs to start from the ground up, through the leg, core and into the arm.
So what should happen straight right is the punch goes out with the elbow in, only turning over at the end of the punch so that the elbow never flares out to the side. You should “sit” down into the punch and drive from the back leg, the lower you are, the more you can use the ground and your leg for power. If you are too upright you can’t get that support from your lower body.
Good punching takes time, and it takes patience. Whenever you change a technique from what you are doing you often lose power and it feels awkward for a little while, but better technique means a higher plateau in the end. It also means slowing down and correcting the form as a regular part of development, as oppose to always trying to throw for power.