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The bore axis of a firearm is the axis which goes through the center of the barrel. In a rifled barrel the projectile will spin around the bore axis as it goes through the barrel.
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Bore to Sight Distance
It is normally desirable to keep the sights of a firearm as close to the bore as possible, since a firearm with a large bore to sight distance will require more sight adjustment when shifting between shooting at different targets at close ranges. On the other hand, a firearm with a short bore to sight distance will need less sight adjustment when changing between targets at close ranges. At longer ranges the bore to sight distance will be of less importance, since gravity has affected the projectile so much that the height difference between the bore and the sight has less to say in total, compared to the bullet drop.
Recoil Control
In pistols the recoil from the fired cartridge (and the slide movement) will make the muzzle rise after each shot, and how high the bore axis sits in relationship to the hand will affect the degree of muzzle rise. A low bore axis will make most of the recoil go directly into the hand, and thus give less muzzle rise. Among other things that affect muzzle rise and felt recoil is the weight of the pistol frame and tuning of recoil springs.