MOA stands for Minute Of Arc (though some in the firearm industry call it Minute of Angle). Let's go back to your middle-school geometry classes and recall that a circle is divided into 360 degrees. If we want to measure angles smaller than a degree, we divide 1 degree into 60 minutes (and if we want to go even smaller, we divide 1 minute into 60 seconds). Therefore a circle is 360 degrees or 360*60 = 21,600 Minutes of Arc (MOA). So what does this have to do with firearms accuracy, you ask?
Well, if you have a circle that is 100 yards in radius, the length of one minute of arc at this distance works out to approximately 1 inch (to be precise, it is closer to 1.047 inches). How do we get this value, you ask? Here's the arithmetic behind it:
Radius of circle = r = 100 yards.
We know that the circumference of a circle = 2 * pi * r.
Taking pi = 3.1415927 approximately, we have circumference = 2 * 3.1415927 * 100 = 628.31854 yards
Now we know that 1 yard = 3 feet and 1 foot = 12 inches.
Therefore, circumference of the circle in inches = 628.31854 * 3 * 12 = 22619.46744 inches
Now, we know that a circle has 360 degrees or 21600 minutes of arc.
Therefore, length of 1 minute of arc = circumference / 21600 = 22619.46744 / 21600 = 1.04719756667 inches.
Since target ranges are usually set in multiples of 100 yards, this makes the measurement rather convenient for shooters. For example, if the firearm is shooting about 3 inches to the right of dead center at 100 yards, then we simply need to adjust the sights 3 MOA to the left to make it hit dead center. For greater ranges, we simply scale up the measurements as required: e.g. for 200 yard range, 1 MOA = 2 inches approximately, for 300 yard range, 1 MOA = 3 inches approximately and so on. Most modern telescopic sights are set to be adjustable in 1/2, 1/4 or 1/8th MOA per click, so it makes zeroing the sights very easy, since we know that 1 MOA = approximately 1 inch for 100 yards distance. Quite a few telescopic sights come with an MOA scale printed around the adjustment knobs.
Telescopic sight which measures in 1/4 MOA increments per click of the adjustment knobs.
As we mentioned above, strictly speaking, 1 MOA is about 1.047 inches for 100 yards. Therefore, some people define a separate term called SMOA (Shooters Minute of Arc) which is defined as exactly 1 inch for 100 yards. Some scopes come with an SMOA scale rather than an MOA scale. The difference between true MOA and SMOA is pretty small: for 1 MOA at 1000 yards range, true MOA works out to be 10.47 inches and SMOA works out to be 10 inches, therefore the difference between the two is less than 1/2 inch for 1000 yards distance. However, if one was to make adjustments of say 20 MOA, then at 1000 yards, the difference between the two would be around 9.4 inches! So it is good to know which unit the scope is calibrated to.
So when someone mentions that his/her firearm shoots 1 MOA, that means that under ideal conditions (no wind blowing, match-grade ammunition used, firearm mounted on a bench rest, barrel and chamber are clean etc.) the firearm will shoot groups of bullets inside a 1 inch circle on average at 100 yards distance. Most quality rifle manufacturers will guarantee that their rifles shoot sub-MOA groups with specific ammunition brands. A sub-MOA means that the rifle will shoot groups of bullets in a circle smaller than 1 inch at 100 yards range. With really high end rifles with match grade barrels and quality ammunition, 0.2-0.5 MOA or better is easily achievable. US Army sniper rifle standards from 1988 require the rifles to shoot a 5 shot group with 0.605 MOA accuracy over 300 yards distance when using M118 special ball cartridges and a government approved bench rest. This works out to shooting 5 shots inside a 1.9 inch circle at 300 yards. Any rifles that fail to meet this standard are returned back to the manufacturer.
To give the reader some idea of accuracy of various firearms, a typical assault rifle shoots about 3-6 MOA, a typical sniper rifle (depending on whether it is used by police or military) shoots about 0.25-2 MOA, but a real accurate competition rifle may easily shoot 0.15-0.3 MOA groups. With new advances in metallurgy and machining techniques, several manufacturers are now offering civilian rifles that are guaranteed to shoot 1 MOA or better out of the box, and cost less than $1000 too. Just a few years ago, such accuracy at such a low price would have been unthinkable. People used to pay their gunsmiths hundreds of dollars to bed their stocks and fit precisely machined custom barrels, all in order to get 1 MOA accuracy from their rifles. Now, due to advances in technology, they can now achieve the same accuracy or better from off-the-shelf guns and ammunition.