I am new to weapon optics and am wondering how eye relief works. For example, say an ACOG has a quoted specification of two inches of eye relief. Does that mean the user's eye must be exactly two inches away from the scope in order to be able to use it? Is there any vertical or horizontal leeway once the eye is positioned at the proper eye relief distance? I am particularly interested in knowing how much eye position variance is allowed while using an ACOG.

Eye relief is the distance you place the optic from your eye. There is not much in the way of leeway for position left or right or up or down of line of sight from the lens. Because the optic is magnified, the image will be distorted if you are not looking straight through the scope. The effect is not as bad on a non-magnified optic like an Aimpoint or EOTech. But, on an ACOG, you want to be looking straight through the scope.

Quoted: I am new to weapon optics and am wondering how eye relief works. For example, say an ACOG has a quoted specification of two inches of eye relief. Does that mean the user's eye must be exactly two inches away from the scope in order to be able to use it? Is there any vertical or horizontal leeway once the eye is positioned at the proper eye relief distance? I am particularly interested in knowing how much eye position variance is allowed while using an ACOG.

It means 2" from scope exit lense to eye is ideal.

The eye should be directly in line with the scope.

Variance is something that usually isn't rated. I have handled scopes that you must be EXACTLY the right distance from them or they don't work right.

I have a compact ACOG and an ACOG-NSN. I would guesstimate the ACOG-NSN gives about 1/2" +/- and still gives a good image. The compact ACOG probably gives more leeway than that.

Would eye relief also be the correct term for the large area in which I am able to see the dot on my open hologram site?

Red dots have unlimited eye relief.  The exit pupil will determine how much up/down/left/right variance there is but not the front/back variance.

Parallax is a shift of the reticle to a spot that is not the same as the POI.  If your scope is not focsued at the distance you are viewing the head position will shift the reticle a lot.  If it is in propper focus this is less of an issue.  ACOGs are calibrated so the reticle ticks are in focus at the distances shown.  This shift is also why there are scopes with adjustbale focus.  Other scopes have fixed focus and are paralax free at a fixed distance like 150 yards usually and head position must be perfectly centered at all other ranges.