MOA is a measurement for the accuracy of a gun, and stands for "minutes of arc" (or minute of accuracy or minute of angle; there is some confusion on this point). In any case, minute does not refer to time, but to degrees; an accuracy of one minute of arc means that the weapon does not deviate from the target by more than 1/60 of a degree; at 100 yards, that is equivalent to a deviation of an inch.
While MOA seems to provide a nice, clean cut evaluation of the accuracy of a weapon, it is in many ways a highly subjective measurement, and affected by (at least) the following factors:
- The weapon: Obviously the weapon itself has a vast effect on the MOA. A sniper rifle would be considered useless if it didn't have an MOA of at least 1/2, most rifles are between 1 and 5, and many pistols and muzzle loaders have an MOA of around 5, though there are certainly exceptions; some muzzle loaders and pistols have been found to have an sub-1 MOA, while a US Army issue M16 can (in theory, at least) have an MOA up to 4 before it is considered unsuitable for use.
- The distance: MOA is, nominally, a measurement independent of the distance, but obviously that doesn't really make sense. As a stupidly extreme example, a 1 MOA gun cannot hit Sol from Earth, even if its MOA might suggest it could easily make that shot. A more practical example is that while a .22 pistol might be quite accurate at 50 yards, the small weight and relatively light charge will mean that by the time it reaches 1000 yards, gravity and air movement will have changed its final resting place quite dramatically. For this reason, MOA is sometimes specified with a distance, such as "1 MOA at 200 yards".
- The ammunition: It is generally agreed that expertly done handloads provide better MOA than factory produced ammunition, probably due to better consistency. In addition, a 'hotter' round with more powder behind it will tend to have a better MOA; there is effectively no difference between the ammunition in an M16 (.223) and in a backwoods varmit huntin' .22, except that the M-16 has a whole lot more powder (and thus, velocity).
- The shooter: I would argue that this is the most important factor in the overall MOA. A skilled shooter can compensate for wind, distance, and movement much better than an untrained and unskilled user, and thus can produce far more accurate results at longer distances.
Note that I have no real idea what I'm talking about (this is primarly based on book lernin and my relatively limited experience with firearms), so any corrections or additions would be welcome.