The health of an in-game character is a way to show it's well-being. When a character is hurt, it's health will lower. If you run out of health, you will be wasted. This means you pass out and need to be revived at at a spawn point, often at the local hospital.
Lost health can be regained in several ways. The most obvious way is to pick up a health icon, which are spread around the map of most GTA's. Other ways are eating, saving the game or picking up a prostitute (note that these options may not work in all GTA's). You can protect your health by wearing armor.
Health isn't shown in the first Grand Theft Auto and GTA London: getting hit by a bullet means instant death. Health was introduced in GTA2, where it was represented by five hearts. From this GTA onwards, health has been shown in the HUD as hit points (percentage), a health bar or a ring around the radar. Health has become less important since GTA III, as you have unlimited lives from this GTA onwards, meaning being wasted would have the same consequences as being busted.