There are several different ways that health is managed in video games. The basic premise is always the same: enough damage can kill you. However, they differ in how much damage can kill you and whether that health can regenerate. So here is my description/analysis of the different models that I have observed. I'm mostly going to be talking about health models in shooters/action games, since RPGs tend to mostly use the first model that I will discuss.
The most classic model of health in video games is the finite, non-regenerative health model. Basically, the hero/heroine has a fixed, initial amount of health (typically 100) and it decreases as s/he takes damage. When health reaches 0, the character dies. Healing is possible through health packs or healing spells if it's an RPG.
This model has been with us since the beginning of video games and is the most familiar. Your health is usually denoted by a number in part of the screen. A shield which can extend your health past 100 is not unfamiliar. One thing about designing around this type of health is that your game becomes boiled down to trying to make it from health pack to health pack, since each health pack resets you back to full health.
One of the downsides of this model is that it can promote Rambo-ing every thing, because taking your time can cause you to take more damage overall.
In this model, you health regenerates back to full. There are two subtypes of this model: slow-regenerative and fast-regenerative. Slow-regenerative typically isn't used. In fast-regenerative models, you must go for some period of time without taking damage before the regeneration will kick in. Also, health totals are smaller in this model than in non-regenerative models. So, a burst of damage will kill the character, making the action more tense, but the game isn't made immensely difficult because you health will come back. This model has no health packs.
A popular, recent example of a game that uses this model is Uncharted 2. Also, games that use this model don't need an on-screen number to indicate health level and will use visual cues to indicate health levels. In Uncharted 2, when you take damage, the screen shifts to gray-scale representing a loss of vision because Drake is dying.
This model, and other regenerative models, encourages safe play and hiding behind cover.
Then there are health models that stride the line between no regeneration and full regeneration.
This is the Halo model. Your character has a shield in addition to his/her health. The shield regenerates, but the health doesn't. So if you take a small amount of damage, it will all regenerate back. If the damage goes through your shield, only the shield will regenerate. This is a direct compromise between full and no regeneration. This promotes safe play, but doesn't let you exploit the fact that all of your health can regenerate.
This model is a little more difficult to explain. Essentially, you have a health level, and a health cap. Whenever you take damage your health goes down, and your health cap goes down less. When you for a period of time without taking damage, your health will regenerate, but only up to the health cap. If you pick up a health pack, your health cap and health will go back up to full. This model was used in Kill Switch for the XBox and PS2 and in many fighting games. In this model, there is no way to regenerate back up to full, so all damage you take has an effect and a sufficiently large burst can kill you.
Every one of these models has its own place, and no one model is better than all of the others in all situations. RPGs like to use the no regeneration model whereas shooters and action games will tend towards one of the other models.