The "Karate punch", launched from the hip in a straight line is perhaps the most fundamental thing we have in our art.  It looks different from the boxer's punch, and yet in truth it has many similarities.  Prounouncing the term for punch has led to it be anglicised in writing as either tsuki or zuki.  You can understand how these are identical, and one should be mindful to try to get the "t" sound in there.

Training while stood upright, feet one shoulder-width apart must be understood to be an exercise.  We would probably not fight with our feet parrallel, but the exercise serves a purpose.  Our punch finds its best use when it has bodyweight behind it.  In fact, the physics of the straight punch and the necessity of keeping the hips and shoulders square rely on the use of the hips to generate power.  That may be a pulse from hanmi to shomen or it may be tucking the tail-bone as we step.
Letting the shoulder run free when practising in thin air is not good for the joint, so muscle contraction and "packing" the shoulder down is used to stop over rotation and over extention.  This is often identified as kime.  In launching the punch, the elbow must be kept alongside the body for the longest portion of time - meaning it does not rotate out to the side.  At the end of the punch, the elbow should still be aimed at the floor.  If it dies anything else then it is hooking.
Hook punches are fine, if that is what we seek to do.  If you are trying for a straight punch then don't hook it.
The connection of the shoulder to the legs is through the abdominal and pectoral muscles on the front of the body and the lattisimus dorsii on the rear.  Although it seems to be about the arm, the act of punching should connect the structure of the body by concentrating on these areas as well as the supporting/generating muscles of the legs.  The Japanese idea of using the hara to generate power requires us to link out punches through our abdomen.
Connecting with the fore-finger knuckle and the middle-finger knuckle is usually termed seiken.
Some style rotate the fist so that the palm faces the floor at full extention.
Some styles rotate the fist so that the palm faces the side at full extention (this is usually termed tatezuki).
Some styles rotate the fist through a three-quarter turn.
We should not let the idea of how far to rotate the fist dominate our thoughts.  The rotation occurs at the end of the punch, and if we are connecting between 50% and 80% of the way through then we will strike the target before we have achieved very much rotation at all.  Of course, if we are connecting at the end of the punch in kumite it will be to prevent harm to our partner as the energy of the punch will already be spent.

The standing punch is often referred to as chokuzuki.  The walking punch is oizuki in Shotokan 
松濤館 .  Punching opposite the lead leg is called gyakuzuki 逆突き. 

Hip rotation is taught while standing in some lineages, and not in others.  As standing punches are just training exercises then whether hip movement is used will depend on what is being trained.  How much of the motion is being trained in that particular exercise.  As it isn't the full range of motion anyway it is just a matter of what degree is being "allowed".

The idea of the zuki is not quite the same as the Western tradition of punching.  Rather, it is seen as a straight thrust which just happens to connect with the fist.  The image that the kanji creates is one of something tubular. 

While hips and shoulders square is the rule, there are plenty of exceptions.  The usual one is the kizamizuki, or jab.  The hips are twisted for a jab.  The usual hip rotation for uke waza (and, in fact, anything that isn't a punch), is to turn the rear hip away, thus rotating with the front hip as a hinge.  The rotation for kizamizuki is moved to the rear hip, thus projecting the front hip and pushing the knee forward.  In practice we must be careful not to let the knee spin inwards.  In Wadoryu
和道流 the jab features a lunge forward with the front foot and is termed junzuki.
Other punches with twisting hips include the uppercut, urazuki, and the "whirlwind punch" furizuki.

hanmi - half facing
shomen - square on
kime - focus
hara - belly
seiken - forefist
tatezuki - vertical punch (not upwards, just a vertical fist rather than a horizontal one).
kumite - matching hands (sparring).
chokuzuki - straight punch
oizuki - lunge punch
Shotokan - Shoto's Hall (Hall of Pine Waves)
gyakuzuki - reverse punch
kanji - calligraphy invented by the Chinese and included in Japanese written language
kizamizuki - jab
uke waza - receiving techniques
Wadoryu - the Way of Harmony school
junzuki - lunge punch
urazuki - back of the thrust
furizuki - storm thrust.
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