Hemorrhaging is bleeding shown by blood pooling on the surface, within the organ, or just simple bleeding from any wounds. Regular hemorrhaging gradually does vital damage when left untreated, while internal hemorrhaging will threaten to burst and do vital damage.
For small hemorrhages, shown by small clouds of blood, antibiotic gel needs to be applied to the affected areas to prevent the hemorrhage from worsening and forming a blood pool. Other hemorrhages shown by blood pools will need to be drained. Note that the blood pools will often prevent you from doing things like suturing cuts or excising foreign objects.
Sometimes blood pools within the organ. If left untreated they will burst, dealing vital damage to the patient. The internal hemorrhages need to be found with the ultrasound, then an incision with the scalpel is needed. Note that this will cause some vital damage. Then, the blood pool needs to be drained, and the incision sutured.
The procedure in Endoscopy is simpler as all that's required is to cauterize the hemorrhage with hemostatic forceps, along with draining the blood pools that form around the wound. Cease cauterizing once the bleeding stops for the most points - if you cauterize for too long it will deal damage to the patient's vitals.
Drain blood pools as quickly as you can - more blood pools will form faster if there is one nearby. This tactic is usually required to stand a good chance of getting the "Less than X blood pools formed" bonus in some operations.
When dealing with internal hemorrhages, pay attention to the size of their shadows when they are viewed with the ultrasound. Larger shadows mean that the hemorrhages are closer to bursting.
When multiple internal hemorrhages are present, and vitals are sufficiently high, you can cut them all out and treat a few hemorrhages at a time, raising vitals in between treatments. Only a certain number of hemorrhages can be present on the organ, so this will lower the risk of one bursting.