While most body modification procedures do involve some blood loss, in most it is minimal and should require no first aid treatment let alone concern on any larger level. However, there are times when blood loss can be significant, both intentionally and unexpectedly.
However, blood loss will result in unconsciousness after loss of less than a liter. Whilst unconscious you will be unaware of further blood loss and could easily lose sufficient blood to result in severe hypotension and hypoxia. Both are potentially fatal if untreated.
A drop in blood pressure will reduce perfusion of the brain, which is the cause of loss of consciousness. Reduction in perfusion of the brain for more than five minutes can result in irreversible brain damage.
Simple direct pressure can stop the bleeding a majority of the time, or pressure on the nearest supplying artery. Elevation above heart-height and a cold compress may also help with swelling. Heavy bleeding that does not stop after this may require medical treatment (i.e. constant fresh flow of blood that cannot be stopped, not a large dermal punched piercing that drips over a week, or a PA piercing that bleeds for the first few days while sleeping.)
In the event of extensive bleeding, pressure should be supplied to the artery at a point between the wound and the heart. This will prevent further blood loss. A tourniquet may be used, but should only be applied for a short time (ten - fifteen minutes and loosened at regular intervals). In these cases, seek immediate medical assistance.
People with bleeding disorders (Hemophilia, etc.) will bleed more, of course, and since their blood can't properly clot, their bleeding should be taken more seriously.