There are only four boxing punches: jab, cross, uppercut, and hook. They are used separately and in punch combinations.
My aunts expressed their concern at me boxing, but one of my uncles told me, "Remember, keep the jab going!" The jab is the bread and butter of boxing. It can be effective all by itself, but it it works marvelously to set up other combinations.
It is thrown from the shoulder. Extend the arm out like a corkscrew. The thumb and palm should be facing down when the fist reaches its destination. Act like a rubber band is on that arm and snap it back to the guard position after delivering it. Don't ever leave it out too long, or else the other person will take advantage of the opening to get in a jab or cross of their own.
The entire body weight is behind this one, so put your all into it. It is usually thrown from your chin with your rear hand (the hand that you're not using as your jab hand). Turn your back foot and your hip as the arm is extended out. Just like the jab, quickly bring your hand back into the guard position.
A boxer can be ambidextrous with this one. This is a good for catching the other person under the chin or in the body. This is traditionally thrown at close range. Shift a little to the right (or the left, depending on which hand is being used to deliver it), and bend the knees a little. The punch hand is thrown upwards in an arc. Push up off of the foot and turn your hips slightly in the direction of the punch to put extra "umph" behind it.
Here's another one that depends on body weight to make it more effective. This is used to catch someone in their jaw, and to break down the sides of the body. Turn the foot as if putting out a cigarette on the ground and turn the hip in the direction of the punch. The elbow is bent as the fist comes in horizontally to get the job done. While this is being thrown, make sure the other hand is up in guard position, protecting the face.
A combination is boxing punches being thrown in succession. Sometimes, one punch being thrown is not enough to score points.
The most famous combination is the old one-two-three. Punches are numbered.
When coaches work with their boxers on drills, instead of calling out the name of the punch, they will name the number of the punch. A jab is one, a cross is two. A left hook is three, and a right hook is four. A left uppercut is five, and a right uppercut is six. So the one-two-three is jab, cross, left hook (or right hook, if your jab hand is also the right hand).
There are many, many combinations. Once you learn the basic punches, add combinations.
There's the overhand right, sometimes referred to as a bolo punch. The right cross is thrown, but at an arc, so that the punch comes down on an opponent. Be careful with this one, because if it is not done correctly, a coach or referee may count it as a slap. Also, this punch leaves a boxer wide open for a right cross or right jab from the other person.
Punches to the kidneys -- usually done via hooks -- are illegal because they can cause some real internal damage. A guy once told me about using the washroom after experiencing one of those punches, and peeing out blood.
Never hit someone in the back of the head or in their back, because that's illegal to do so as well.
And here are the boxing basics.