I figured I would post this as a refresher if your a bit rusty or for others who may not know to learn and be able to sight in very quickly and effectively this is my procedure others are also correct this is just what I find works well to sight in with a fairly minimal amount of rounds
MOA , Minute of Angle is an angular measurement
looking at a compass you see 360 degrees each degree of angle is then broken up in to Minutes for a smaller measurement
like an minute of time a Minute of angle is 1/60th of the degree
a 60th of a degree or a Minute of Angle MOA is approximately 1 inch at 100 yards it is actually 1.047 inches which only really mattes beyond 500 yards so just 1 inch at 100 yards for sight in purposes
in this diagram you can see that beyond an MOA being and inch at 100 yards 2 inches at 200 yards 3 inches at 300 and so on , the fractions also work 50 yards is 1/2 inch and 25 yards is 1/4 inch .
I also included 12.5 yards 1/8th of 100 yards a MOA is 1/8 inch because sometimes you get that gun that is not on a sheet of paper at 25 yards , if you can move the target in to 12.5 yards 37.5 feet you don't need to be exact pace it of is fine it is about getting on paper to find out what adjustment you need
now if you can't get on a 20x20 inch target at 25 yards you likely have other mounting issues
as an example a Vortex crossfireII 2-7x32 has 60moa of windage and 60moa of elevation adjustment if you are greater than 15 inches off point of aim with point of impact it is a good time to check your mounts.
you also want the scope as close to the center of it's adjustment range some people when they mount a new scope will turn the adjustment all the way right then all the way left counting stop to stop then back to the middle and the same for elevation adjustment this centers the cross hair in the middle of it's adjustment range where you would like for it to be for best function. if all the mounting is strait this should get you a good starting point to fire your first group from 25 yards.
Inches = Minutes = Clicks
at 25 yards a MOA is 1/4 of an inch on the target most common hunting rifle scopes are 1/4 moa per click of adjustment
If you are off 2 inches at 25 yards you are off 8 moa and to move your optic at 4 clicks per moa you need to turn 8moa X 4 clicks per moa=32 clicks in the appropriate direction.
this is a link to my preferred sight in target for 25 yards because it makes it simple
https://appleseedinfo.org/smf/index.php?topic=9833.msg83223#msg83223 the one inch squares with grid or the one inch squares with grid ink saver
this is 1 inch black squares on a 1/4 inch graph paper , because 1moa = 1/4 of an inch at 25 yards each grid square is 1MOA meaning all you need to do is count the grid squares you need to move over from the center of the group to the center of where you aimed
in this target I fired 3 rounds with my red marker , find the center of the group , then count over squares to my point of aim and count up squares to my point of aim , the center black square (this is the ink saver version) I counted 8 right and 6 up.
I need to move 8moa right x 4 clicks per MOA = 32 clicks , most newer scopes like the Vortex have numbers around the dial and not just clicks you can more easily move 8moa without counting clicks.
notice the green dot that is where I really want to be having bullet impact with most center-fire rifles at 25 yards because the optic is typically about 2 inches above the bore axis so when calculating generally you want impact 1 inch low at 25 yards so that you are not excessively high when you move to your 100 yard target
because I want to be 1 inch below the center of the point of Aim at 25 yards I only want to move 2 moa up 2moa x 4 clicks = 8 clicks up
once I dialed in my correction , I will tap the housing of the turret with side of my thumb knuckle , this is old habit and it may not be needed any longer but it doesn't hurt , back 30-40 years ago it was not uncommon to dial in a correction only to have that correction actually take place during the recoil of the next 1-3 rounds , when I used to work gun deer sight in's I learned from the old timers there who had little brass hammers around 1 ounce they would dial in correction and 5-6 taps with the little brass hammer on the turret housing and it would help some of those scopes to move before the shot. I made up a set of tiny brass hammers for my dad and I when we worked sight ins but I frequently forget to bring it to the range on one or two rifles my thumb knuckle works fine
then I fire a second group typically it is almost exactly where I intended it if my group is within an inch of intended it is good enough as long as it isn't above the center of the point of aim 1 inch higher than the top of the square would be 6 minutes high at 100 yards and since that is compounded by still being below line of sight at 25 yards it is more like 10MOA high that could easily be off the target at 100 yards.
if the group moves the wrong direction farther away from point of aim by the amount you moved you moved the adjustment the wrong direction go the opposite direction twice this is where is is very handy to have made notes of your clicks and direction with arrows indicating the rotation you dialed in.
if the group strings toward the point of aim you may want to fire a few more rounds till it settles to a group.
some low power and red dot optics are 1/2moa adjustment , the manual should indicate this in that case number of MOA needed x 2 clicks per MOA = correction