Range has two components: damage drop-off over range and arrow flight trajectory. The higher the range a bow possesses, the more damage it will do to far-away targets and the straighter the arrow will fly, making it easier to hit targets at a distance, too.
Certain bows, such as the Composite Bow, possess very poor range, limiting their threat value to enemies that are fairly close. Other bows, such as the Black Bow of Pharis, have superior range ratings and can be used to effectively snipe distant enemies. Bows with poor range tend to hit below and slightly to the left of targets (demonstrating the Archer's paradox), with the effect becoming more dramatic the further away the target is. Notably, damage will be drastically reduced to the point of being negligible, although enemies will still be aggroed. At extreme ranges, enemies will not aggro even when hit, although the damage is also negligible at this point. Enemies will still turn towards arrows striking hard surfaces, regardless of range.
Wearing the Hawk Ring increases the range of any equipped bows.