Yep, scopes are their own arcane science. First the nomenclature...
Scopes are typically described with something like 2-7x28mm. The numbers in front of the "x" show the magnification level of the scope. If there is only one number it is a fixed power scope. If there are two numbers in front of the "x" separated by a dash it is a variable power scope with the numbers giving the lowest and highest magnification levels. So 2-7x28mm is a 2 to 7 power scope with a 28mm objective lens. The ring that adjusts from one magnification level to another is usually at the end of the scope nearest to your eye.
The number to the right of the "x" is the diameter of the objective lens. As a general rule, bigger lenses give you brighter, clearer images. There is a break even point where the extra size adds little. That break even point is about 44mm so you don't need anything wider than that. How much smaller than 44mm it can be is related to the magnification level of the scope because of a factor called "exit pupil."
Exit pupil is what makes a scope "bright" and is a factor in how close you have to get your eye to the scope to see through it. Bigger is better. To figure out how big the exit pupil is, divide the size of the objective lens by the magnifcation level, so a 40mm lens at a mag level of 12 would have an exit pupil of 3.3mm. If your max mag level is only 9 then you can get the same brightness out of a 32mm lens (32 / 9 = 3.56) as the 12 power scope gets out of a 40mm lens. So, by extension, if you want lots of magnification you need a 44mm lens; if you want, say, 9x or less you can happily use a 32mm lens.
The next factor to consider is "parallax." A scope is focused to be parallax free at a given range. For discussion, let's say it is 100 yards. What that means is that the reticle (crosshairs) and a target at 100 yards are both on exactly the same visual plane. If you put the center of the reticle right on the "x" in the center of the target at 100 yards and then move your head around without moving the scope the reticle will stay right on the "x" --. there is no parallax. But if you now put the reticle on the "x" of a target at 50 yards and try the same trick you will see a slight movement in the reticle as you move your head around. If your eye is left of perfect center the reticle will look like it is to the right of the "x," and so on; there is parallax or, more properly, there is a parallax error. If you have parallax error you can't be sure your reticle is in the right place unless you get your head in exactly the same place each and every time you put your eye behind the scope. Obviously, you want to eliminate parallax error. How do we fix this? Two answers.
The first is to get a scope that is focused (parallax free) at the mid-point of the ranges we shoot most often. For a rimfire, say, we shoot at 25 yards to 100 yards so we want the scope focused at about 50 - 75 yards, and that is where rimfire and air rifle scopes are focused to be parallax free. You will still have a very small parallax error but it will be at the extremes of your ranges and not in the middle. Depending on the type of shooting you do this may be perfectly acceptable. Most "big bore" scopes are focused to be parallax free at about 125 yards and will have significant parallax error at the ranges typically used by rimfire shooters.
The other solution is to have a scope that you can focus to the exact range you are shooting at that moment; this is what an "adjustable objective lens" does for you. It is usually just called "adjustable objective" or "AO." Most often, scopes with AO have that designation added to the end of their description, such as 3-9x40mmAO. Most AO scopes use a focusing ring right on the objective lens. If the scope is a fixed power scope the focusing ring is sometimes at the end of the scope nearest the eye. Sometimes it is on the side of the scope.
Big bore scopes with AO usually focus down to 50 yards. Rimfire scopes have to get to much shorter ranges than that; how short a range they focus to is one measure of their quality so you always want to ask what their minimum focus range is. You certainly don't want one that stops at 50 yards if you plan to shoot at 25 yard targets. Most I have seen get down to 10 yards but it is something you have to check.
The next consideration is just how much magnification you need. If you are doing what I call "precision target shooting" at 100 yards you want lots of magnification. What I mean by "precision target" is that you are trying to get five shots into an area smaller than a 1" square or circle. To do this you need enough magnification to know for certain your reticle is precisely on the same aimpoint each and every time you pull the trigger. For this kind of work I would say 12x is a miniumum; 16x is better. I use 24x on my rifle that I use for this kind of shooting and think that is about as high as I need to go.
If you are doing that kind of shooting at 50 yards, with even smaller targets, I would say 9x is the absolute minimum and 12x would be better.
At the other end of the magnification level discussion we have hunting and plinking. Here, too much magnification is a hindrance because of the smaller field of view. High mag levels--> small field of view --> trying to find Mr. Rat while looking at the world through a soda straw. On the other hand, low mag levels--> wide field of view--> using a paper towel roll.
Again, range comes into play. If your hunting targets show up at 50 yards you can probably get by with a 4x scope. If they pop up at 25 yards you probably want a 2x, a "red-dot" scope, or no scope at all. If they are fast movers (rabbits) you need a wide field of view--> low power or red-dot or no scope, whereas stationary targets (squirrels hanging out near the top of a tree, rabbits lounging under bushes at longer distances) can get by with slightly less field of view--> more power.
Next up is the type of reticle. Duplex reticles have a wide crosshair that narrows down to a skinnier crosshair where they cross. But there is skinny and then there is skinny. The standard duplex crosshair will completely hide the X on most targets at 50 and 100 yards, so you don't know if you are aiming at the top of the "x," the middle of it, or at the bottom.
For super precise aiming you want "target crosshairs" or "fine crosshairs" that show you exactly where on the X you are aiming. These hairs look smaller than a pencil line at 100 yards, allowing you to dot the "i" on the fine print at the bottom of the target. But, as usual, there is a tradeoff -- fine crosshairs can disappear in dim light when you are trying to aim at Mr. Rabbit when he is standing against a dark background. Even in good light they are sometimes hard to find in a hurry, and almost impossible to keep sight of when following a moving target through the scope.
Most (but not all) mil-dot reticles use fine crosshairs, as do "target dot" reticles.
Even though it is not the best for precise target shooting, I prefer the standard duplex reticle. Sometimes it frustrates me when I'm going for small groups at 100 yards, but I see it is a good "general purpose" reticle suited to the type of shooting I usually do.
And lastly is the type of adjustment knob you want. Target style knobs are taller, can be turned by hand instead of requiring a coin or screwdriver, and are made to be adjusted frequently. If you move from a 25 yard target to a 100 yard target you just crank the knob the proper amount in the proper direction, put your crosshairs on the center of the target and shoot.
Hunter style knobs are made to be set and forgotten about. You sight in at your most frequent or desired range and if your target changes range you just aim over or under as needed to get the bullet where you want it.
Okay, where does that leave us? Maybe here:
Plinking at the occasional soda can, informal bull's eye matches or "small group" competition at 50 yards now and then, with the occasional long shot at that can 100 yards away? Then 2-7x or 3-9x or 4-12x will do for a variable magnification scope; and 4x or 6x will do if you want a fixed power scope. An objective lens of 28mm - 33mm will do just fine, but 40mm might be better if you go with 4-12x.
Regularly target shooting at ranges out to 100 yards or beyond? Then you'll want 16x or better along with a larger objective lens (40mm or more) to make up for the high mag level.
Shooting mostly at one range in one session? Hunter knobs.
Shooting at many different and very different ranges during one session? Target knobs.
Scopes are typically described with something like 2-7x28mm. The numbers in front of the "x" show the magnification level of the scope. If there is only one number it is a fixed power scope. If there are two numbers in front of the "x" separated by a dash it is a variable power scope with the numbers giving the lowest and highest magnification levels. So 2-7x28mm is a 2 to 7 power scope with a 28mm objective lens. The ring that adjusts from one magnification level to another is usually at the end of the scope nearest to your eye.
The number to the right of the "x" is the diameter of the objective lens. As a general rule, bigger lenses give you brighter, clearer images. There is a break even point where the extra size adds little. That break even point is about 44mm so you don't need anything wider than that. How much smaller than 44mm it can be is related to the magnification level of the scope because of a factor called "exit pupil."
Exit pupil is what makes a scope "bright" and is a factor in how close you have to get your eye to the scope to see through it. Bigger is better. To figure out how big the exit pupil is, divide the size of the objective lens by the magnifcation level, so a 40mm lens at a mag level of 12 would have an exit pupil of 3.3mm. If your max mag level is only 9 then you can get the same brightness out of a 32mm lens (32 / 9 = 3.56) as the 12 power scope gets out of a 40mm lens. So, by extension, if you want lots of magnification you need a 44mm lens; if you want, say, 9x or less you can happily use a 32mm lens.
The next factor to consider is "parallax." A scope is focused to be parallax free at a given range. For discussion, let's say it is 100 yards. What that means is that the reticle (crosshairs) and a target at 100 yards are both on exactly the same visual plane. If you put the center of the reticle right on the "x" in the center of the target at 100 yards and then move your head around without moving the scope the reticle will stay right on the "x" --. there is no parallax. But if you now put the reticle on the "x" of a target at 50 yards and try the same trick you will see a slight movement in the reticle as you move your head around. If your eye is left of perfect center the reticle will look like it is to the right of the "x," and so on; there is parallax or, more properly, there is a parallax error. If you have parallax error you can't be sure your reticle is in the right place unless you get your head in exactly the same place each and every time you put your eye behind the scope. Obviously, you want to eliminate parallax error. How do we fix this? Two answers.
The first is to get a scope that is focused (parallax free) at the mid-point of the ranges we shoot most often. For a rimfire, say, we shoot at 25 yards to 100 yards so we want the scope focused at about 50 - 75 yards, and that is where rimfire and air rifle scopes are focused to be parallax free. You will still have a very small parallax error but it will be at the extremes of your ranges and not in the middle. Depending on the type of shooting you do this may be perfectly acceptable. Most "big bore" scopes are focused to be parallax free at about 125 yards and will have significant parallax error at the ranges typically used by rimfire shooters.
The other solution is to have a scope that you can focus to the exact range you are shooting at that moment; this is what an "adjustable objective lens" does for you. It is usually just called "adjustable objective" or "AO." Most often, scopes with AO have that designation added to the end of their description, such as 3-9x40mmAO. Most AO scopes use a focusing ring right on the objective lens. If the scope is a fixed power scope the focusing ring is sometimes at the end of the scope nearest the eye. Sometimes it is on the side of the scope.
Big bore scopes with AO usually focus down to 50 yards. Rimfire scopes have to get to much shorter ranges than that; how short a range they focus to is one measure of their quality so you always want to ask what their minimum focus range is. You certainly don't want one that stops at 50 yards if you plan to shoot at 25 yard targets. Most I have seen get down to 10 yards but it is something you have to check.
The next consideration is just how much magnification you need. If you are doing what I call "precision target shooting" at 100 yards you want lots of magnification. What I mean by "precision target" is that you are trying to get five shots into an area smaller than a 1" square or circle. To do this you need enough magnification to know for certain your reticle is precisely on the same aimpoint each and every time you pull the trigger. For this kind of work I would say 12x is a miniumum; 16x is better. I use 24x on my rifle that I use for this kind of shooting and think that is about as high as I need to go.
If you are doing that kind of shooting at 50 yards, with even smaller targets, I would say 9x is the absolute minimum and 12x would be better.
At the other end of the magnification level discussion we have hunting and plinking. Here, too much magnification is a hindrance because of the smaller field of view. High mag levels--> small field of view --> trying to find Mr. Rat while looking at the world through a soda straw. On the other hand, low mag levels--> wide field of view--> using a paper towel roll.
Again, range comes into play. If your hunting targets show up at 50 yards you can probably get by with a 4x scope. If they pop up at 25 yards you probably want a 2x, a "red-dot" scope, or no scope at all. If they are fast movers (rabbits) you need a wide field of view--> low power or red-dot or no scope, whereas stationary targets (squirrels hanging out near the top of a tree, rabbits lounging under bushes at longer distances) can get by with slightly less field of view--> more power.
Next up is the type of reticle. Duplex reticles have a wide crosshair that narrows down to a skinnier crosshair where they cross. But there is skinny and then there is skinny. The standard duplex crosshair will completely hide the X on most targets at 50 and 100 yards, so you don't know if you are aiming at the top of the "x," the middle of it, or at the bottom.
For super precise aiming you want "target crosshairs" or "fine crosshairs" that show you exactly where on the X you are aiming. These hairs look smaller than a pencil line at 100 yards, allowing you to dot the "i" on the fine print at the bottom of the target. But, as usual, there is a tradeoff -- fine crosshairs can disappear in dim light when you are trying to aim at Mr. Rabbit when he is standing against a dark background. Even in good light they are sometimes hard to find in a hurry, and almost impossible to keep sight of when following a moving target through the scope.
Most (but not all) mil-dot reticles use fine crosshairs, as do "target dot" reticles.
Even though it is not the best for precise target shooting, I prefer the standard duplex reticle. Sometimes it frustrates me when I'm going for small groups at 100 yards, but I see it is a good "general purpose" reticle suited to the type of shooting I usually do.
And lastly is the type of adjustment knob you want. Target style knobs are taller, can be turned by hand instead of requiring a coin or screwdriver, and are made to be adjusted frequently. If you move from a 25 yard target to a 100 yard target you just crank the knob the proper amount in the proper direction, put your crosshairs on the center of the target and shoot.
Hunter style knobs are made to be set and forgotten about. You sight in at your most frequent or desired range and if your target changes range you just aim over or under as needed to get the bullet where you want it.
Okay, where does that leave us? Maybe here:
Plinking at the occasional soda can, informal bull's eye matches or "small group" competition at 50 yards now and then, with the occasional long shot at that can 100 yards away? Then 2-7x or 3-9x or 4-12x will do for a variable magnification scope; and 4x or 6x will do if you want a fixed power scope. An objective lens of 28mm - 33mm will do just fine, but 40mm might be better if you go with 4-12x.
Regularly target shooting at ranges out to 100 yards or beyond? Then you'll want 16x or better along with a larger objective lens (40mm or more) to make up for the high mag level.
Shooting mostly at one range in one session? Hunter knobs.
Shooting at many different and very different ranges during one session? Target knobs.
