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Damage modifiers refer to the buffs or debuffs applied to a unit that causes it to deal or receive increased or decreased damage. These are percentage values. They don't add extra flat damage but modify all damage normally done.
These buffs or debuffs are received by champion abilities, neutral buffs, on-hit effects from items and runes.
Damage dealt modifier
These are buffs that allow the champion that has them to deal increased or decreased 
magic or 
physical damage. The raw value from the attack or ability is increased or decreased by the modifier and then applied to the target. All damage modifiers 
stack multiplicatively.
Increasing damage dealt
Champion abilities
Ashe's Frost Shot- Aurelion Sol's Celestial Expansion
- Corki's Hextech Munitions
- Hecarim's Rampage
- Jinx's Fishbones
- Karthus' Lay Waste
- Kha'Zix's Taste Their Fear
- Lucian's The Culling
- Miss Fortune's Double Up and Bullet Time
- Mordekaiser's Obliterate
- Nidalee's Javelin Toss and Takedown
- Rek'Sai's Furious Bite
- Renekton's Reign of Anger
- Riven's Wind Slash
- Rumble's Junkyard Titan
- Ryze's Overload
- Shyvana's Twin Bite
- Sion's Decimating Smash, and Unstoppable Onslaught
- Sona's Hymn of Valor
- Syndra's Force of Will
- Tristana's Explosive Charge
- Vayne's Tumble
- Vi's Vault Breaker
- Xerath's Eye of Destruction
- Yasuo's Sweeping Blade
- Ziggs' Short Fuse and Mega Inferno Bomb
Items
Runes
Flat bonus damage
While technically not damage modifiers, these abilities are included for completeness.
Champion abilities
- Ashe's Ranger's Focus
- Gangplank's Powder Keg
- Lucian's Lightslinger
- Miss Fortune's Love Tap
- Skarner's Crystal Slash
Items
- Doran's Shield (Unlike most sources of bonus damage, this is added after damage modifier calculations)
- Doran's Ring (Unlike most sources of bonus damage, this is added after damage modifier calculations)
Decreasing damage dealt
Champion abilities
- Jhin's Whisper
- Kalista's Martial Poise
- Kled's Skaarl the Cowardly Lizard
- Sona's Aria of Perseverance
- Yasuo's Way of the Wanderer
- Yone's Way of the Hunter
Summoner Spell
Damage received modifier
These are buffs or debuffs that allow the champion that have them to receive increased or decreased magic or physical damage from autoattacks, abilities and item's actives. The final value from the attack or ability is increased or decreased by the modifier and then directly applied to the target's health.
Armor and magic resistance damage reduction percentage count as one source of damage received modifier against physical and magical damage respectively. For example, a champion with 300 armor has 75% damage reduction from physical damage.
Damage reduction from armor and magic resistance and from any other sources stack multiplicatively.
Increasing damage received
Champion abilities
Items
Runes
Decreasing damage received
Champion abilities
- Alistar's Unbreakable Will
- Braum's Unbreakable
- Galio's Shield of Durand
- Garen's Courage
- Gragas' Drunken Rage
- Irelia's Defiant Dance
- Jax's Counter Strike (area of effect abilities only)
- Kassadin's Void Stone
- Kled's Dismount
- Malzahar's Void Shift
- Master Yi's Meditate
- Warwick's Primal Howl
Summoner Spells
Items
- Frozen Heart (Basic attacks only)
- Plated Steelcaps
- Randuin's Omen
- Warden's Mail (Basic attacks only)
Damage immunity
There are some instances in the game where a champion is immune to receiving any kind of damage (including true damage) for a few seconds.
Champions
- Bard's Tempered Fate
- Ekko's Chronobreak
- Fiora's Riposte (excluding Turrets)
- Kalista's Fate's Call
- Kayn's Transformation
- Kayle's Divine Judgement
- Kindred's Lamb's Respite
- Lissandra's Frozen Tomb (self-cast only)
- Pantheon's Aegis Assault (excluding Turrets and only invulnerable to damage coming from a specific direction)
- Sion's Glory in Death
- Taric's Cosmic Radiance
- Xin Zhao's Crescent Guard (only invulnerable to champions a certain distance away)
- Zilean's Chronoshift
Items
Flat Damage Reduction
Several champions have abilities that reduce damage by a flat amount. Some flat damage reductions are factored in after armor or magic resistance. This makes it significantly better the more resistances you have. In addition, periodic damage effects will have each tick of damage reduced. For example, 
Malzahar's 
Malefic Visions ticks 8 times, so 1 flat damage reduction would reduce its total damage by 8.
Flat damage reduction does not work against true damage.
