Newtons is a force measurement, specifically one favoured by metric. More generally though force is mass times acceleration so actually lbs/pounds is a measurement of force (the number of pounds you weigh being the force, unlike kilograms which is actually mass if you want to be a glorious pedant. This gets several levels deeper if you wanted as it can vary a slight bit between places but even I will skipn that for the moment. The imperial equivalent of kilograms/mass being a slug if you were curious.). If you take an engineering course this whole manner of thinking is something you might encounter as dimensional analysis which is a super powerful tool (see video at the end).
Pounds per square inch is a measurement of pressure (force divided by area if we are sticking with the dimensional analysis thing), indeed one of the customary ones (force divided by area means you can do something like lbs per rood or newtons per square foot if you wanted). In this case if the area of contact is a square inch (in some ways fairly small area but you might have small hands (I always suspected) or you could be getting really technical and only measuring knuckles or something.
Hitting things can also be a more tricky area to contemplate as many will also be inclined to measure impulse and momentum and consider them the greater part of things -- there is a reason hitting something with a hammer can dislodge things that take many tonnes if you were to use a hydraulic press or something.
https://sciencedemonstrations.fas.harvard.edu/presentations/karate-blow
Choice video on dimensional analysis. Sixty Symbols on measuring the power of a nuclear bomb
