Boresighting a rifle is not strictly a necessary step, but it can save you time and ammunition when sighting in a newly mounted scope or set of iron sights. With a bit of prep work using some ballistics software (or a pencil and paper) you can use two points to boresight a rifle at a very short range, as little as a few yards, so that when you do go to the range, you should already be on the target at a distance of 25 or 50 yards (or meters). Boresighting is only an approximation, so additional sighting in using the desired ammunition under field conditions is still required.

This method of boresighting does not require any special equipment, but it will only work on firearms where you can look down the barrel when the sights are firmly attached. Bolt action rifles, with the bolt removed, are ideal, as are break action firearms with the sights on the barrel. AR rifles will work, if you remove the bolt and lower receiver. On firearms where the bolt is not easily removed, such as lever actions, may be boresighted with this method if you can use a small mirror to look from the breech down the bore, as can revolvers. Recoil operated semiautomatic pistols cannot use this method, as there is generally no way to look down the bore while the barrel is locked into alignment with the slide. You can still boresight this type of firearm, but you will need something like a laser boresighter to establish your bore axis.

Supplies

A way to hold the rifle steady (a rifle rest, a padded vise or Workmate type bench, or a bipod and a wedge)
A target at your initial sight-in range or a vertical surface at your boresighting distance and a couple of sticky notes
A tape measure
A ballistics program (This one can be downloaded and run in a browser) or a pencil and paper

Step 1: Some Ballistics Background

First, a bit of background. Looking through sights or a scope mounted above the barrel, your line of sight can be considered a straight, level line going from the sights to the center of the target. The bullet starts out below the line of sight, and the barrel will be angled slightly upwards (some fraction of a degree, usually). When the bullet leaves the barrel, it will travel upwards as well as forwards, crossing the line of sight. This point is called the near zero, and is the first of potentially two points where your shots should group exactly on the center of the target. As the bullet travels on, it will rise above the light of sight, but due to gravity pulling the bullet down, it will rise more and more slowly until it begins to fall. This curve is the bullet's trajectory, and will be somewhat parabolic in shape. As the bullet falls, it will cross the line of sight again, and that point is called the far zero, and it is the second point where your shots should group exactly in the center of the target. In general, the near zero will be anywhere from 10 to 50 yards, and the far zero can be at the same point as the near zero, our out a thousand yards or more. In what may be the reverse of what many people expect, the closer the near zero, the further out the far zero; this is because a close near zero means more barrel elevation, which means more vertical travel before the bullet begins to fall.

Boresighting adds another line to this, the bore axis. This is a line that runs down the center of the barrel, parallel to the bore. At the moment the bullet exits the barrel, it is travelling down the bore axis, but as soon as it is free of the bore, it starts to fall along the trajectory. For a short distance, the trajectory will be close to the bore axis, and that fact is what makes boresighting work.

If you have a ballistics chart (either provided by the ammunition manufacturer, reloading manual, or ballistics software) it should give you values for the rise and fall at different ranges. If the table says, for instance, that your bullet should be 1 inch low at 10 yards, then mark a sheet of paper with two marks an inch apart. You will want to use the shortest range listed, so that the trajectory and bore axis are as close as possible. The IdealZero program listed in the supplies section will not only give you the trajectory values, but will calculate the difference between the bore axis and the line of sight at 5 and 10 yards, allowing you to boresight with more precision.
If you're just going to wing it without a ballistics chart, then you'll want to boresight at a reasonable near zero distance. For a slow bullet, like a .22 Long Rifle, a pistol caliber carbine, or something similar, 25 yards is probably a good distance. For a fast bullet 2000 fps or more, 50 yards makes more sense. In this case, since you are boresighting at the zero range, you only need a single mark, so just use the center of a target. Expect your first shots to land a few inches below the center of the target.

If you wish to use the IdealZero software, follow the link https://github.com/fluzwup/IdealZero and clink on the green "Code" button. Select the "Download ZIP" option. Extract the contents of the zip file into a directory, then point your web browser at "IdealZero.html". Enter the details of your cartridge and firearm and click the "Click to run" button. Be honest with the dispersion; if you tell it you can shoot a 0.5 MOA group, but you can really only shoot 5 MOA groups under the conditions you'll be shooting, such as out hunting, then you will get a zero that gives you a 50% chance of missing at the mid-range trajectory. Giving good numbers will get you a zero that will maximize the maximum point blank range (MPBR). Note that the fact that this runs locally from a web browser means you can put the software on a phone, tablet, or laptop, and take it to the range with you, with no internet connection required.

Step 2: Setting Up the Rifle

A loose gun can never be sighted in, so make sure that every component is secure before you start the process. Check the tightness of all the screws:

Action into the stock
Barrel into the receiver, if applicable (such as Ruger rimfires with slip-in barrels)
Scope bases to the gun
Scope rings to the bases
Scope rings around the scope

Do not over-tighten, because that can cause damage, such as crushed stocks or stripped screws; make sure everything is tightened and apply a non-permanent thread locker where appropriate. This will help prevent the screws from working loose under the shock of recoil. It can even help to lightly tap on solid parts, like scope bases and rings (but not the scope itself!) with a plastic or wooden mallet, just to make sure everything is firmly seated and won't shift.

Next set up the rifle securely in the rest. An adjustable rifle rest is ideal if you have it, but you can get by with a bipod or some sandbags to hold the forend, and a wedge (a 2" binder works well) to allow you to move the butt of the rifle up and down in very small amounts. However you secure the rifle, it is important that the rifle stay vertical, so that the line of sight is directly above the bore axis. You can then either adjust the angle of the rifle by moving the butt up and down (easiest for one person), or adjust the target up and down (this is best with two people, since one needs to be looking down the bore).
Now set your boresighting target up at the appropriate distance from the muzzle of the rifle. Adjust the rifle and/or the target so that the appropriate aiming point (the lower if you have to points) is visible and centered in the bore of the rifle. Once this is done, make sure the rifle and target do not move by double-checking periodically during the sight adjustments.

Step 3: Adjusting the Sights

Now we should have the bore axis set just how we want it, so we need to carefully adjust the sight line to the appropriate spot. In the image above, you can see the two red arrows through the scope, which is set to its lowest power to provide the largest field of view and best focus at short range. The scope has already been mostly adjusted, the crosshair needs to move slightly down and left to be centered on the tip of the upper arrow. Once adjusted, the bottom arrow should point to the center of the bore axis; if not, the rifle needs to be moved to get the bore axis centered on its mark, and then the scope adjusted to center the crosshairs on the line of sight mark.

One thing to be aware of when doing this process with a scope is parallax. Parallax is the difference between the apparent distance to the crosshairs and the distance to the target. If you have an adjustable objective scope, set the adjustment to the actual distance to the boresighting target. When the objective lens is correctly adjusted, moving your eye around should result in the crosshairs staying steady on the target. If the parallax is not set for the range to the target (rimfire scopes are often set to 50-75 yards or meters, centerfire for 100-150) then the crosshairs will move relative to the target as you move your eye around. To eliminate the parallax error, you need to keep the crosshairs centered in the image, as shown in the image above.

Step 4: Take It to the Range

And the final step is to go to the range and try it out. Start at close range, such as 25 yards, and fire a few rounds. They should be close to where you expected for that distance. In my case, I was hitting about 2 inches low at 25 yards, and I should have been 1 inch low. A quick adjustment to the vertical, and I was hitting where I wanted to. Moving out to 50 yards, the rifle was grouping right at the center of the target.