Basic combat rules:
To attack, roll 1d20 +/- any modifiers. If the result is higher than the target’s Armor Class (AC) you hit and do damage. If it equals or is less than the target’s Armor Class (AC) you miss.
When you hit, you do damage. Damage subtracts from the target’s Hit Points (HP). The damage depends on what you are attacking with, various stats, occasional class abilities or monster powers. As long as something has Hit Points (HP) they have avoided any serious damage. Losing Hit Points (HP) means you get a shallow cut, or just exhausted from blocking a blow, bruises, or other minor injuries that will be fine after a few minutes rest and basic medical attention.
If you roll a natural 1 (the number on the d20 is 1) you miss, regardless of modifiers or target Armor Class. If you roll a natural 20 (the number on the d20 is 20) you hit, regardless of modifiers or target AC. A natural 20 is also considered a critical hit. The attacker may either do full damage without rolling -or- pull of some fancy move like trip their opponent, disarm them, shove them back, stabbed the monster in the eye, or whatever else the Player wishes to pull off (within reason). This is in addition to the normal rolled damage.
The three Player Character attributes that have the most bearing on combat rolls are Strength (STR), Dexterity (DEX), and Constitution (CON).
Dexterity (DEX) bonus is added to a Player Character’s attack roll.
Strength (STR) bonus is added to a Player Character’s damage roll.
Constitution (CON) bonus reduces the Hit Point (HP) damage a Player Character takes.
Damage, Death, and Dismemberment.
Damage done is subtracted from Hit Points (HP). As long as a Player Character (PC) has Hit Points they can avoid actual physical damage from attacks and dangers. So if a character has even 1HP they are not actually damaged beyond a few scrapes and minor bruises. Any attack that does more damage than remaining HP just reduces the target’s HP to 0. For most monsters and NPCs this means death.
For PCs when they are reduced to 0 HP, they must make a Constitution check to stay conscious.
Any damage taken when at 0 HP gets a roll on a Death and Dismemberment Table that has all sorts of nasty things that happens.
Initiative and combat timescale:
Combat is measured in Combat Rounds. Each Player Character (PC), Non-Player Character (NPC), Monster, or other entity/etc in the fight gets to do one thing a round. Move, Attack, get something out of a backpack, etc.
Combat Rounds are only a few seconds long so each round happens all at once. So everyone plans what they will do during the turn and the Dungeon Master (DM) works out when everything goes. If there is a need to see which action gets pulled off first (such as two people fighting in melee with each other) then an Initiative Check is rolled by the Player Character (PC) involved. This is a d20 roll where the Player Character goes first if it rolls under the Player Character’s Dexterity and the enemy goes first if that roll is over the Player Character’s Dexterity.
Somewhat better explanation here: (where I got the idea) (Except I narrowed it down to a single action per turn, you can move or attack, not both, in a single Round)
The result is that for each Round, every PC announces what they are doing each Round and the DM adjudicates and resolves the actions in whatever order seems to fit the situation, until two actions directly conflict with each other or it is vitally important which action happens before another one.
The DM should give a general course of action for his side for the Player Character’s can plan their action. Should be general and something visible. “The bandits are determined to keep fighting” or “The kobolds are barely holding their pointy sticks and look ready to flee” sort of thing.
Hopefully this will encourage faster paced and more organic combats that don’t have Players planning out multi-action rounds with bonus actions, full actions, move actions, reactions, and so on. Also hope that it will lead to less waiting around for each Player’s turn to do something to come around. Player Characters should all work together and the action/response cycle should be much shorter.
A few rare supernaturally fast monsters will always win the Initiative Check, while many slow monsters will automatically lose it. The vast majority will be in the same human-scale as Player Characters.