Updated Answer as of Gen 9
The following abilities grant Pokemon immunities, whether it is from specific moves, abilities, or status conditions:
Armor Tail/Dazzling/Queenly Majesty: Gives Pokemon with one of these abilities and their allies an immunity to priority moves
Aroma Veil: Gives Pokemon with this ability and its allies an immunity to moves/abilities that limit its move choices (Taunt, Torment, Disable, Encore, Heal Block, infatuation, Cursed Body. Note that Psychic Noise will only deal damage and not activate the Heal Block effect it has when used on a Pokemon with Aroma Veil and its allies)
Big Pecks: Gives Pokemon with this ability an immunity to Defense lowering moves
Bulletproof: Gives Pokemon with this ability an immunity to ball and bomb moves
Comatose: Pokemon cannot be statused and is considered to be asleep
Clear Body/Full Metal Body/White Smoke: Prevents stat drops from moves and Abilities
Dry Skin/Water Absorb: Restores 1/4 HP when hit by a Water move (Note that Dry Skin can also restore 1/8 HP in rain and lose 1/8 HP in sun as well as taking more damage from Fire moves)
Earth Eater: Restores 1/4 HP when hit by a Ground move
Flash Fire: Boosts the power of Fire moves when hit by one
Flower Veil: Grass-type allies are immune to status conditions and stat drops
Good as Gold: Pokemon is unaffected by Pokemon's status moves
Guard Dog/Suction Cups: Pokemon cannot be forced to switch/flee by moves such as Roar, Circle Throw, Whirlwind, or the item Red Card
Hyper Cutter: Pokemon with this ability cannot have their Attack stat lowered by moves or Abilities
Immunity/Pastel Veil: Pokemon (and its allies for Pastel Veil) cannot be poisoned
Inner Focus/Own Tempo/Oblivious/Scrappy: Immune to the Ability Intimidate (Gen 8 onwards). Own Tempo prevents Pokemon from being confused, and Oblivious prevents Pokemon from being infatuated and being affected by Taunt (Gen 6 onwards for Taunt)
Insomnia/Vital Spirit/Sweet Veil: Pokemon (and its allies for Sweet Veil) cannot be put to sleep
Keen Eye/Illuminate: Pokemon's accuracy cannot be lowered (Illuminate has this effect starting in version 2.0.1 of Pokemon Scarlet/Violet)
Leaf Guard: Pokemon with this Ability cannot be statused during sunny weather
Levitate: Pokemon with this Ability cannot be hit by Ground-type moves
Lightning Rod: Gets a +1 Special Attack boost when hit by an Electric move
Limber: Pokemon with this Ability cannot be paralyzed
Magic Guard: Pokemon with this ability do not take damage from indirect damage (i.e Leech Seed, Toxic)
Magma Armor: Pokemon with this Ability cannot be frozen
Motor Drive: Gets a +1 Speed boost when hit by an Electric move
Overcoat: Pokemon with this ability are immune to sandstorm and hail damage, powder and spore moves (i.e Sleep Powder, Poison Powder, Stun Spore), and the Ability Effect Spore starting in Gen 6
Purifying Salt: Pokemon cannot be statused
Sap Sipper: Gets a +1 Attack boost when hit by a Grass move
Shields Down: Pokemon cannot be statused when it is above 50% HP
Soundproof/Cacophony: Pokemon are immune to sound moves
Storm Drain: Gets a +1 Special Attack when hit by a Water move
Sturdy: Provides an immunity to OHKO moves
Thermal Exchange/Water Bubble/Water Veil: Pokemon with any of these abilities cannot be burned
Volt Absorb: Restores 1/4 HP when hit by an Electric move
Well-Baked Body: Gets a +2 Defense boost when hit by a Fire move
Wind Rider: Pokemon with this Ability are immune to wind moves (i.e Heat Wave, Icy Wind, Hurricane, Blizzard)
Wonder Guard: Immune to direct attacks that aren't super effective
Source