The harm that your character inflicts in an attack is determined by the number of successes that you get with your attack roll. Each success inflicts one Health point of damage.
Example: You roll five dice for your character to stab an opponent and get 1, 4, 8, 8 and 9. You get three successes. Your character's opponent therefore loses three Health points (See your sheet for this information).
The type of damage done in a successful attack is determined by the weapon used or the nature of the attack.
If you get no successes on your attack roll, your character does no harm to his target. The attack misses altogether or is ineffectual.
After you determine the damage that your character inflicts upon her target, the Storyteller portrays the damage in descriptive terms, narrating the outcome of the attack. Rather than simply say, “Okay, the guy loses four Health,” the Storyteller makes events interesting. He might announce, “You plunge your knife into the guy's dead flesh and pull upward, wrenching until the blade hits bone. The bastard screams in agony, but there's no blood. You watch as his chest collapses and begins to decay at an accelerated rate.” By being evocative, your Storyteller creates atmosphere, entertaining you and lending a sense of narrative continuity to what would otherwise be a series of dice rolls.
See: Health and Healing Times for more information on how damage affects your character.