Health is an abstract measure of your character's resilience expressed in numerical form. In other words, it's how much damage characters can take before expiring. It's a straight-forward statistic, depleted by taking damage (in crises or through dialogue choices), and restored by resting or using healing items, like the sprayflesh.
When Health is fully depleted, the Last Castoff dies, but death is not always the end. Instead, you will usually revive in the Calm and be able to resume your travels upon returning to the real world.
Companions you take with you on your travels do not die when their Health is depleted fully. Instead, they are knocked unconscious and will come to once it is safe, retaining a tiny amount of Health.