When sighting in a gun scope, what does left and up mean?

I have a Nikon prostaff and I'm not sure about sighting it in. I did it last year, but I don't want to waste ammo by being wrong. So, what goes left when you turn the dial left is it the corsshairs? Or the image? Also, is it the same with up and down?

And! If it moves 1/4th inch at 100 yards, that would be 1/2 inch at 50, right?

Left and Up are the directions the bullet will move on the target. Left to move the bullet impact to the left, etc. Yes, that would be 1/2 inch at 50. You might want to read the owner's manual for your scope.

Edit: That would be 1/8th inch at 50 yards.

The center of the scope is the bull's-eye. The up & down terms are what you have to do to the bullet strike point to get it to the bull's-eye. You sight the scope to the bull's-eye & fire a round. The impact point of that round is below the bull's-eye, so you have to move the scope UP to get the strike to the bull's-eye.

If you want the hits on the target to go higher, turn in the direction indicating up on the elevation knob.
If you are hitting too far left on the target, turn in the direction indicating right on the windage knob.

The cross hairs move when you turn the turret knobs. But, it's imperceptible when you're looking though the scope and making small changes. If your gun was locked down in a rest, you could probably tell if you did a several MOA adjustment. At any rate, since you're keeping the cross hairs on the target, that relationship doesn't change, so in effect, you are moving the barrel in the direction of the change (changing the alignment off the barrel and the scope.)

When you crank the up knob in the up direction, it means that the bullet will hit higher in relation to the cross hairs. When you crank the left button in the non-left direction, it means that the bullet will stay in the chamber (not leave).

The directions indicate the direction than the point of impact will move. So if the bullet hits lower than where you're aiming then you'd turn the turret toward "up" to move the point of impact up.

And if it's 1/4 inch at 100 yards then it would only be 1/8th at 50 yards. It changes by an angular amount… The distance between two lines at an angle to each other will always be larger further away, not smaller.