In most classic adventure games, a character has some number of Hit Dice, often (though not always) gaining one each time they gain a level. Traditionally, only the new dice are rolled, meaning that a character who rolls poorly will always be disadvantaged, while one who rolls well will always have some advantage.
Instead, I suggest that each time a character gains an additional Hit Die, whether by gaining a level, or by some other means, the player should roll their entire pool of Hit Dice, adding any modifiers due to the result. If the result exceeds their current maximum Hit Points, then the maximum increases to this level. Otherwise, their maximum Hit Points remain unchanged. Their actual Hit Points do not increase automatically; they must rest (as below) to gain the full benefit of their increased resilience.
In this way, some level of regression to the mean will occur. A player who rolls poorly at first is likely to see a marked increase when they gain a level. A player who rolls well may see little or no improvement. But there is always some chance of either outcome. And in the long run, feeble wizards with low Constitution scores and small hit dice are all but guaranteed to be more vulnerable to harm than mighty warriors with high Constitution and large hit dice.
When a character takes damage to their Hit Points, this always reflects some actual, though trifling, wound which saps the character's fortitude. It will be seen from the increasing numbers of Hit Points as characters gain levels that they are increasingly unconcerned by such trifles. Nonetheless, each blow landed offers an opportunity for acid, poison or the like to take effect.
When a player character is reduced to 0 Hit Points, they are not yet dead - though at this point every blow becomes potentially mortal, as discussed below. If using Critical Hits - however you choose to define them - these may have bypass Hit Points entirely and have a chance of directly inflicting a mortal wound. This offers the opportunity for a deadlier, more exciting game.
Monsters and non-player characters will generally die (or at least be out of the game) at 0 Hit Points. The Referee may choose to allow dragons, important non-player characters, and the likes to roll for injuries.
Injuries
When a character reaches 0 HP, roll on the Injury Table with a penalty equal to the damage in excess of that required to reduce them to 0 HP. Some results on the table do not automatically result in the character being removed from combat - especially if their foe has marked them for death. In this case, each subsequent hit requires another roll on the Injury Table, with a penalty equal to the full amount of damage received. It will be obvious that surviving a blow once at 0 HP is unlikely - though possible, if the character has luck.
Injury Table (Roll 2d6)
1 or Less: Grisly Death. Body so spectacularly destroyed that only a resurrection or wish spell can bring it back to life
2: Instant Death: Decapitated, or similarly grievous wound. No hope of survival.
3: Mortal Wound. Disemboweled, stabbed through the heart, etc. Dead in 2d6 rounds. Save vs. Death or unconscious until death.
4: Slow Death. Internal bleeding, punctured lung, or other similar. Dead in 2d6 turns. Save vs. Death or unconscious until death.
5: Lose a limb: Roll to determine limb affected. If head rolled and no helmet worn, treat as Instant Death. If body rolled and no armour worn, treat as Mortal Wound, otherwise dead in 3d6 turns from blood loss. Takes 3d6 weeks to recover from wounds. Save vs. Death or unconscious until death.6: Paralysed: Devastating blow damages your spinal cord. Roll to determine loss of muscle and sensory function. 1-3: legs; 4-5, waist down; 6, whole body. Save vs. Death or unconscious for 3d6 turns.
7: Broken bone: Roll to determine location. 3d4 weeks to heal. Save vs. Death or unconscious for 3d6 rounds. If head is struck, no helm is worn, and save is failed, unconscious in coma until healed.
8: Knocked Out: Unconscious for 2d6 rounds and helmet destroyed. If not wearing a helmet, treat as Broken bone: head.
9: Stunned: Cannot take actions for 1d4 rounds and helmet destroyed. If not wearing a helmet, knocked out for 2d6 rounds instead.
10: Flesh Wound: Ouch. This one hurt. Cannot take actions for 1 round. If not wearing armour, cannot take actions for 1d4 rounds.
11: No Effect: You got off lightly this time.
12 or more: Adrenaline Surge: You gain 1d4 hit points per hit die. At the end of combat, you lose these hit points, and suffer a -1 penalty to all rolls. This penalty increased by a further -1 for each round of combat in which you act, and reduces by 1 for each turn you completely rest.
Hit Location (d6): 1, left arm; 2, right arm; 3, left leg; 4, right leg; 5, torso; 6, head.
Clearly, wearing a helm is a good idea. Why wouldn't you? That's a matter for when I talk about armour. But in short, can't see well, can't hear well, look like you're after a fight. Not getting hit is even better, which is why Shields Shall Be Splintered.
Other Death and Dismemberment tables are available. I like the 2d6 ones; this one owes a lot to Robert Fisher, Trollsmyth, and Troll and Flame. But make your own to suit your preferred level of carnage. Or add modifiers to the roll: Constitution makes sense.
Players should track each wound received by their characters, along with the time taken for them to heal. While a character is wounded, the character's recovery of HP is reduced by 1 HP per night of rest per wound, to a minimum of zero. If a character has the misfortune to suffer multiple serious injuries, the recovery times are not cumulative: wounds will recover in parallel.
I've intentionally not listed ways of treating wounds. An imaginative party can find ways to do it; the Referee should adjudicate them reasonably and fairly.
Recovery
HP is recovered at a rate of 1 HP per night of sleep, generally comprising six to eight hours in one or two stretches, possibly with a period of keeping watch. In a normal low-level party of four fools with swords adventurers, eight hours will allow one person to keep watch while the other three sleep.
If you are able to get a good night's rest - undisturbed, well-fed, warm, and in a proper bed - this increases by 1 HP per night. This will usually require that you spend the night in someone's home, an inn, or something of that nature, though a sufficiently wealthy party may be able to achieve this in the wilderness. Any requirement to keep watch or subsist on iron rations rules out a good night's rest. Note that in the common room of an inn, undisturbed sleep cannot be guaranteed!
If you receive attentive care, this increases by a further 1 HP/night. A nurse can only care for one injured person at a time, and is unable to rest while they do so. In extremis, one party member might nurse another, at the expense of their own rest and recuperation, though normally it will be necessary to seek aid elsewhere. This allows for recovery of a maximum of 3 HP/night. At the Referee's discretion, higher rates of recovery might be possible with the attention of a skilled physician, though this should come with a significant cost to the character's purse and at some risk to their health.
Those who are quick with numbers will note that if a character has somehow accumulated 3 wounds and not died yet, they will likely incapable of recovering hit points at all until their wounds heal. This isn't a system suitable for a 'save the world before next Tuesday' campaign. Long gaps between adventures, or stables of characters, are going to be essential.
Magical Healing
Healing magics, however administered, knit wounds together quickly and crudely. They may be quite sufficient to mend minor cuts and scrapes, though the crudeness of magic for this purpose means that scarring is almost inevitable.
In game terms, this means that the likes of cure light wounds are able to restore hit points. But they cannot cure Injuries received via rolling on the status table. A spell such as regenerate might do the trick, though at a cost - for serious wounds, healing magic is really quite inadequate to the task.
These spells will merely reattach flesh and bone in the position where they lie. In this case, scarring is inevitable, and worse deformations are possible if (for example) a broken bone is reset while the limb is contorted. The character may, at the referee's discretion, continue to suffer some of the detriments of their injury - for instance, a badly-set broken arm may still be unable to wield a weapon.
In this case, a true cure, if possible, will require the attention of a surgeon or physician with the skill to safely operate on the casualty. This is likely to involve inflicting fresh wounds on the character: breaking a badly-set bone, for example. And that, in turn, means more time out of action, waiting for the wound to heal, and potentially catching some nasty disease.
No, magic might be a useful way to stay alive long enough to reach proper medical care. But that's really all it is. Go visit your doctor, folks.