To hit a punch shot, the golfer sets up with the golf ball farther back in his stance, takes a shorter backswing, and makes an abbreviated follow through. A punch shot is typically played with a longer club than a normal swing would require from the same distance (for example, if you would normally use a 7-iron, you might choose 6-iron or 5-iron to hit a punch).
Why hit a punch shot? The most common reason is keep the ball from ballooning up into a strong swing - either a headwind or crosswind. In windy conditions, you can keep more control over the golf ball (and maintain or gain distance) by keeping the ball on a lower trajectory. A punch shot helps you do that. A punch might also be played to keep the ball underneath tree branches, or in any other situation where you want to hit the ball lower.
A "punch shot" and a "knockdown shot" are essentially the same thing for recreational golfers. Highly skilled golfers might think of them as distinct shots because golfers with great shotmaking ability can make minute adjustments in technique. Problem is, if you ask such golfers to describe the difference between a punch and a knockdown, you'll get many different (and sometimes contradictory) answers. So recreational golfers should just think of "punch shot" and "knockdown shot" as interchangeable terms.
See also:
How to hit a punch shot
Quick tip from Fuzzy: the knockdown shot
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"That tree might be in the way, you should probably punch this shot underneath those branches."
"He's going to have to punch this one to keep the ball down."

