52 Abilities, three different ways to change the result
The damage calculator lets you pick an Ability from the list the selected Pokémon actually has, not from the full table of 306 entries. Of those, 52 actually change the number in the result, and they do it in three different ways: some reduce damage to zero, some multiply the power of a particular kind of move, and some change Defense or Attack itself before the formula is even calculated. The distinction isn't cosmetic. Confusing "weaker" with "zero" changes the answer to whether a given move is worth using at all.
Full immunities: zero, not a quarter
Eleven Abilities turn a specific move type into a total immunity: Levitate against Ground moves, Water Absorb, Storm Drain, and Dry Skin against Water, Volt Absorb, Lightning Rod, and Motor Drive against Electric, Flash Fire and Well-Baked Body against Fire, Sap Sipper against Grass, and Earth Eater against Ground. In every case the damage drops to zero, but what happens instead varies from Ability to Ability: Water Absorb and Volt Absorb restore a quarter of max HP, Storm Drain and Lightning Rod raise Sp. Atk, Motor Drive raises Speed, Flash Fire powers up the defender's own Fire moves, and Well-Baked Body sharply raises Defense. So the same "immunity" can be a punishment for the attacker or a gift for the defender. The calculator only counts the damage dealt, not the side effect on the opponent's side.
Two more Abilities ward off not a type but a specific property of a move: Bulletproof blocks ball and bomb moves (Shadow Ball, Focus Blast, Energy Ball, and twenty-two other moves), and Soundproof blocks sound-based moves (Boomburst, Hyper Voice, Overdrive). Wonder Guard goes further than either of them: it lets through only super-effective moves. Everything else, even a clean neutral hit, deals zero.
Abilities that boost one type or one condition
Four Abilities boost moves of one specific type, whether or not the Pokémon itself is that type: Steelworker powers up Steel moves, Dragon's Maw Dragon moves, and Rocky Payload Rock moves, each by half, while Transistor boosts Electric moves by 30% (the Gen IX version; in Gen VIII it was by half, so the number depends on which era of the game you have in mind). Adaptability doesn't boost a type; it raises the same-type attack bonus (STAB) from ×1.5 to ×2, so it only works when the Pokémon is attacking with its own type anyway.
Seven more respond to a condition other than type. Huge Power and Pure Power double physical Attack unconditionally. It's the simplest rule on the whole list, but also one of the strongest, because it multiplies the stat itself rather than the final damage. Technician boosts moves with 60 power or less by half, meaning weak moves, not strong ones, the opposite of what intuition might suggest. Tinted Lens doubles not-very-effective moves (×0.5 becomes ×1, the same as a neutral hit), and Neuroforce boosts super-effective moves by 25%: two Abilities answering opposite questions about type effectiveness. Sheer Force boosts moves with an added effect (a chance of a status condition or a stat drop) by 30%, but in exchange that effect stops working. It's the only Ability on the list that takes something away in return for what it gives. Solar Power raises Sp. Atk by half in the sun, at the cost of an eighth of max HP every turn in a real battle. The calculator only counts damage dealt, so that cost doesn't show up here.
Starter Abilities: the power kicks in only at low HP
Overgrow, Blaze, Torrent, and Swarm all have the same shape: when HP drops below a third of the maximum, moves of the Pokémon's own type (Grass, Fire, Water, and Bug, respectively) get half again as strong. The set of types isn't a coincidence. Grass, Fire, and Water are the types of the starter trio in almost every game, and Bug usually goes to the first Pokémon you catch right after leaving town. An Ability that only switches on when the Pokémon is close to fainting is a last-ditch desperation Ability, not one for opening a battle. The calculator lets you pick it regardless of the HP you enter, so the result assumes the condition is met rather than checking it for you.
Abilities that depend on a move's property, not its type
Four Abilities don't care what type a move is, only what property it has, and properties are often less obvious than type. Iron Fist boosts punching moves by 20% (21 moves in the site's database, from Mach Punch to Plasma Fists). Tough Claws boosts contact moves by 30%. Most physical moves make contact, but not all of them: Earthquake is physical yet doesn't make contact, so Tough Claws doesn't touch it. Strong Jaw boosts biting moves by half (ten moves, including Crunch and Fire Fang). Mega Launcher boosts aura and pulse moves by half (Aura Sphere, Dark Pulse, Dragon Pulse, Water Pulse). A fifth Ability from this family, Reckless, doesn't ask about the move's property but about recoil: it boosts the eleven moves that cause recoil, from Take Down to Head Smash, without reducing the recoil itself. Rock Head does the opposite in the game (it removes recoil instead of boosting the move), but you won't find it on this list: since it doesn't change damage, the calculator has nothing to multiply here.
Abilities that weaken hits without zeroing them
A few Abilities on the defending side work more gently than a full immunity: the damage goes down but doesn't disappear. Thick Fat halves both Fire and Ice moves at once, while Heatproof halves only Fire. Purifying Salt halves Ghost moves; in the game it also protects against status conditions, which the calculator doesn't account for. Filter, Solid Rock, and Prism Armor are three different Abilities with an identical effect: super-effective moves are a quarter weaker, so ×1.5 instead of ×2 against a single type advantage. Multiscale and Shadow Shield, also two Abilities with one effect, halve damage, but only at full HP. The calculator always calculates the first hit on a healthy Pokémon, so here the condition is always met. Ice Scales halves only special moves, and Fur Coat doubles Defense against physical moves: two Abilities, two different move categories, the same effect at a glance.
When two rules meet in one move
A few Abilities have two separate rules at once, and what happens when both apply to the same move isn't always obvious. Fluffy halves contact moves but doubles Fire moves. A Fire move that also makes contact (Flare Blitz, Fire Punch) hits both rules at once and the multipliers cancel out: the result is exactly what you'd get with neither. Punk Rock does opposite things with the same kind of move on each side: on the attacker it boosts its own sound-based moves by 30%, and on the defender it halves incoming ones. Water Bubble is the only Ability on the list that does something completely different on each side: on the attacker it doubles its own Water moves, and on the defender it halves Fire damage. Two unrelated rules that just happen to share a name.
What the calculator deliberately leaves out
Not every Ability that changes damage in the game has a counterpart on this list; a few depend on something the calculator doesn't model at all. Intimidate lowers the opponent's Attack by one stage, but there's no entry for it in the Ability list. You set the same effect manually in the stat stages field. Sharpness and Wind Rider ask about move properties ("slicing" and "wind") that our dataset doesn't include, and guessing them from move names would produce numbers that look certain. Weather Abilities like Drought and Drizzle deliberately didn't get their own entries: you set the weather directly in the weather field, and duplicating the same effect in two places on the form would multiply questions, not information.
Four Abilities that care about how the turn plays out
Guts and Marvel Scale respond to status conditions, and the status of both sides is a separate field on the form, explained in the guide to weather, terrain, and status in battle. The calculator really does account for them: Guts turns the burn penalty into an Attack bonus, and Marvel Scale raises a statused Pokémon's Defense by half against physical moves. Analytic asks about move order within the turn, and Stakeout asks whether the opponent has just come onto the field. A single exchange of hits doesn't capture that, so both get a number with the assumption spelled out right next to the result. That's a deliberate choice: leaving Stakeout out would make the result wrong by half in exactly the situation this Ability exists for.
Test it on your own pair of Pokémon
All 52 Abilities are calculated in the damage calculator along with nature, IVs, EVs, held items, weather, terrain, stat stages, screens, and status, and the form always limits your choice to the Abilities the selected Pokémon actually has. Levitate won't show up as an option for Charizard, because Charizard doesn't have it. To see which circumstance really decides a matchup, the fastest way is to enter a specific pair and compare the numbers instead of multiplying in your head.