Battery acid contains concentrated H2SO4 (sulphuric acid). The acid is a powerful dehydrating agent so it can burn the skin and affect layers under the skin. There is also a lot of heat so it can cause blisters/burns It burn in cotton cloths and make holes.
Extent of damage will depend on concentration of acid, duration of contact, quantity of acid, area in contact, part of body ( some areas are more delicate hence more damage).