One minute of angle (think of it as a very thin slice of pie) subtends an arc of ~ 1.0472" at 100 yd. The arc is the circle (piecrust) and the 100 yd range is the radius of the circle. If your gun shoots with one minute of angle precision, you should be able to put all your shots into a circle on your target that is equal to 1.0472" in diameter. One half minute of angle would be all the shots in a circle about 1/2" in diameter. As the range increases (radius of the circle), the arc length (piecrust) increases proportionally. So at 500 yd, a 1 MOA shot group would be equal to ~ 5 x 1.0472" (~ 5 1/4"), or five times larger than at 100 yd. Sub-MOA simply means a group at some distance that was in a circlular diameter less than one minute of angle.
Of course, the more shots you fire, the more chance you'll pull one, or the wind will blow it off line and the group will get bigger. Generally, groups of less than five shots are considered suspect when used to imply some level of precision for a given rifle. 5 shot groups are a little better indicator, and 10 even better. There are plenty of statistical analyses for group sizes versus shot number if you care to search. As the number of shots in a group increases, at some point you are more likely measuring the precision of the shooter, rather than the rifle. In any case, take a look at the cartoon below, which is a representation of the pie analogy I used above in MOA and milliradians (mils), which are just a slightly larger angle or slice of pie. Hope that helps answer your question.
As another example, the targets below were shot at 300 yd, 15 shots per target. The inner circle scratched into the surface (around the red dot) is ~ 0.45 minutes of angle, in other words, a 1.4" diameter circle at 300 yd. The second circle is ~ 0.9 MOA, or 2.85" in diameter at 300 yd, and the third is ~ 1.9 MOA (5.85" in diameter). These correspond to the scoring rings on a 300 yd Reduced Palma F-Class target. As you can see, all the shots on Targets 1 and 2 are either touching, or inside the 1 MOA line. Therefore, they would be one minute groups. On Target 3, which actually was 17 shots, I pulled a couple to the right. So this is more like a 1.5 MOA group.
