Dragons in Dungeons & Dragons are Colour-Coded for Your Convenience: each tincture is assigned its own special flavour of Breath Weapon. Red, Brass, and Gold = Fire, Blue and Bronze = Lightning, Green = Poison/Corrosive Gas, Black and Copper = Acid, White and Silver = Freezing Air. Most of its imitators (including NetHack) follow suit, though different games do not always follow the same color-to-damage assignments as the original (though red is more likely to remain fire than any other).
Note that metallic dragons have TWO breath weapons (usually some kind of nonlethal but disabling gas attack: Brass have sleep gas, Bronze have repulsion gas, Copper have slow gas, Gold have Fortitude-reducing gas, Silver have paralysis gas; and one straight-up damage attack) and that there are tons of REALLY WEIRD dragons (Iron dragons breathe molten iron, Mercury Dragons LASER BEAMS, some gem dragons breathe EXPLODING CRYSTALS...)
Dragon- based classes give a breath weapon. The most classical example are dragon shaman and dragonfire adept.
Wing Dragons have the only breathweapon that OSHA (the Occupational Safety & Health Administration) would probably approve as harmless. It is sky painting smoke... that they can telekinetically control... and turn optic black... and then blind their targets by surrounding them in... and then do stuff like slitting the throats of in the dark.
Beige Dragons have an invisible line of pure elemental ennui that makes you stupider and less interesting as a person.
Dungeons & Dragons, et. al., offer a substantial number of nondraconic monsters with their own Breath Weapon: Hellhounds breathe fire, as just one example. A popular method of making a "new" monster is to simply slap a Breath Weapon on an existing animal or mythical creature: the Pyrohydra, for instance, is a Hydra that breathes fire!
Also, certain 'metabreath' feats can allow all these creatures to breathe something different, or even shape their breath. Often abused when one type of dragon is impersonating another...
4th edition dragon breath weapons are technically vomit weapons. The edition also adds the Dragonborn as a core Player Character race, who possess their namesakes' signature attack.
More specifically, the Draconomicon teaches that the magical energy of a D&D dragon's breath weapon is stored in the stomach, not in the lungs; this is what is meant by "vomit weapon". The expelled effect is explicitly magical, though—it is mystical energy, and not a biological by-product as found in some other D&D creatures.
The Greater Drakes in 2nd and 3rd edition have non-magical breath weapons; most drakes collect or generate a substance in their throat pouch that they can spit at enemies (excatly what it is varies by species; for three examples: the Smoke Drake generates thick smoke, the Muck Drake stores the swampy water of its lair, and the Hive Drake has a nest of hornets living in its pouch).
Erfworld's dwagons; at least the red, green, blue, purple, and brown ones; have normal elemental breath weapons (fire, poison gas, lightning, sonic blast, and smoke respectively) while pink ones breath pink bubbles that smother enemies, while yellow ones inverse the trope by having massive bowel movements. Yellow dragons can give you a really crappy day.
Played with in Gold Digger. Sliding up the scale, Iron and Copper dragons have fairly generic elemental breath weapons, usually fire. Golden dragons are renowned for having completely random breath weapons, which are affected by their magical auras as well. At the top of the peak, Platinum dragons don't have breath weapons at all, but rather "ether vents", which are essentially small points on their body that can channel destructive levels of magic effortlessly (when used for attack, Word of God has compared them to Star Trek ship phasers.)
Warhammer, as Dungeons and Dragon's punk-rock cousin, the breath weapons tend to be grottier too. Trolls have a breath weapon/vomit attack where they can squirt out a glob of stomach acid so strong it can melt steel. The worst by far are the Great Unclean Ones, greater daemons of Nurgle, god of decay. The Great Unclean Ones can open their mouths to unleash a torrent of fecal matter, mucous, garbage and maggots. As bad as that is, the crap they squirt out is magical and full of the essence of decay. So any victim of this attack will either rot away to nothing or mutate into a daemoon.
Warhammer 40K: In addition to the delightful Nurglite spells mentioned above, that Orks have a spell called Psychic Vomit. You can guess which of the caster's orifices it comes from. Some Tyranid creatures have bio-plasma, where they hork up balls of superheated gas from their stomachs that's set on fire by clicking internal armor plates to create sparks.
The ability is so common and iconic, in fact, that when a player says a creature "has firebreathing" it's immediately understood by most experienced players exactly what that means, word-for-word. It's also a fairly powerful ability as well.
The developers seem reticent to actually make "firebreathing" a keyword, so its not actually the official name of the ability (the ability doesn't have a name). Probably because they are reluctant to keyword anything that's color-specific.
The game has the dragon template, which of course includes a Breath Weapon.
The various Breathe Ice/Fire/Radiation/Steam spells from GURPS: Magic.
The Dresden Files' RPG has Breath Weapon as one of the available powers. The description broadens it so that the power can cover any self-generated projectile, even if it doesn't come from the mouth. This notably includes the Shen demons' use of flaming poo projectiles.
Most dragon enemies in Talisman have a breath weapon, which they use on attacking characters before they enter combat. There is usually a condition that determines whether the character is affected by the breath attack, be it decided randomly with a die roll, based on a character having certain types of equipment in their possession (like weapons or armor), or based on the character's strength or craft scores. While many of the breath attacks cause characters to lose additional life, others may cause them to miss a turn, lose their spells, discard their fate, or anything in between. Some dragon breath weapons even affect other cards or characters that are unfortunate enough to share the same board space as the dragon.
The D&D-descended 13th Age has these for dragons, as expected, but sorcerers have a variation known as the breath weapon spells - Breath of the White, the Green, the Blue, the Black, and the Grave (because the Red ain't much for sharing). The breath weapon rule means that while the spells can usually only be cast once per day, after you cast them, you get a roll at the beginning of each action you take for the rest of the battle to see if you get another use, although you can only roll for one breath weapon at a time. A couple of their Talents, namely Chromatic Destroyer Heritage and Metallic Protector Heritage, make breath weapon spells more powerful, encouraging sorcerers who take those talents to grow into endless volleys of dragonbreath.
Arduin RPG, The Compleat Arduin Book 2: Resources
Fire Demons can breathe out a cone of violet flame 30 feet long and 10 feet wide (at the base) once every three melee rounds. The flame does 36 Hit Points of damage and has a 75% chance of igniting flammable materials.
Ice demons can exhale a freezing cloud that blinds all inside it and does 17-36 Hit Points of cold damage.
Night Demons can breathe a 15 foot long and 3 foot wide tongue of black flame that causes 60 Hit Points of damage and permanently takes away two points of Constitution.
The Greater Demon Amon-Rha can breathe out a cloud of poisonous gas in a cone 60 feet long and 30 feet wide at the base up to eight times per day. The cloud lasts 5-10 melee rounds and does 11-20 Hit Points of damage per melee round.
13 times per day the Greater Demon Boak Chaos Hoof can breathe out a 45 foot diameter cloud of chlorine gas that does 21-40 Hit Points of damage and causes any victim with less than 30 Hit Points to choke to death after one melee round of exposure.
Dagonus the Death Dragon. This three-headed Greater Demon can generate a 180 foot long lightning bolt or 90 foot long cone of flame 7 times per day per head.
The Greater Demon Nagandas, Lord of the Sea Serpents, can breathe out jets of high pressure water that do 21-40 Hit Points of damage.
The Greater Demon Shabaleth, the Dread Elf. Once every five melee rounds he can breathe dragon-like flames in a cone 20 yards long and 20 feet wide at the base that do 31-50 Hit Points damage.
The Greater Demon Urandos, the Blizzard Queen. Ten times per day (but only once every five melee rounds) she can breathe out a cone of cold 60 feet long and 30 feet wide at the end. It does 31-50 Hit Points of damage.
The Tarakk is a giant horned lizard/dragon that can breath out a cone of fire 90 feet long and 15 feet wide at the end that inflicts damage equal to its own Hit Points. The fire breath can be used once every three melee rounds, up to four times per day.
Varanae generic RPG supplement Monstrum 1. Eurytion are a race of giants that can breathe out a flame attack up to 30 yards away once per minute. The flame does an amount of damage equal to the Eurytion's Hit Points.
Chaosium's supplement All the Worlds' Monsters Volume III
The Baggol has a breath weapon that does 4-24 Hit Points damage with the creature's choice of side effect: sleep, fear, fire or acid.
Balitorr, King of the Earth Elementals, has a breath attack that can turn opponents to stone, usable twice per day.
Chestnut/Pearl: its breath weapon is the same as the Dungeons & DragonsWeb spell but lasts indefinitely. Its breath can also be used in reverse to destroy all types of webs.
Hazel/Jade: its breath inflicts paralysis on all creatures in a 70 foot long by 30 feet wide (at the base) cone.
Tan/Turquoise: breathes out flame in a 90 foot long by 30 feet wide (at the base) cone.
The Fire Lion's fiery breath does 8-48 Hit Points of damage to any creature it hits.
The Frost Horror is an elephant-sized white dragon with red eyes. It has a fiery breath that does 13-78 Hit Points of damage in a cone shape 60 foot long and 30 foot wide (at the base).
The Glamdar is a blue-gray monster covered with metallic scales and spines. It can breathe out a cloud of radioactive gas that does 5-50 Hit Points of damage to all inside it.
The Steel Golem has 100 Hit Points and breathes out a 30 foot by 30 foot cloud of chlorine gas three times per day. The entry doesn't say how much damage it does, so assume it's the same as a 1st Edition Dungeons & Dragons dragon and does damage equal to its own Hit Points.
The Hell Sprite is a short two-legged monster with two glowing red eyes. It has a breath weapon that does 2-12 Hit Points of fire damage to all creatures in a cone 10 feet long and 5 feet wide at the base.
The breath of an Ice Mouse does 1-6 Hit Points of damage to a single target.
The breath of a Kendry's Night Demon does 10-80 Hit Points of fire damage.
The Mandadon has the body of a giant snake and the wings and head of a white dragon. Its breath does 10-60 Hit Points of cold damage.
A Mantigon is a cross between a manticore and a blue dragon. It can breathe out a lightning bolt that does 5-60 Hit Points of damage.
The Nasnas is a huge panther-like animal that constantly hunts for and kills dragons and dragon-kin. It has a very useful breath weapon: it negates dragon breath weapons on contact.
The Osta Beast's breath is so foul that any creature breathing it must make a saving throw versus poison or be nauseous for 10-20 minutes and be unable to anything but vomit.
The Scorpan is a human/scorpion crossbreed. It can breathe out a flood of acid that does damage of 4-48 Hit Points.
The Shock Troll can breathe forth a cloud of poison that does 5-30 Hit Points of damage immediately and causes death within 1-20 melee rounds. It fills an area of 1,000 cubic feet and can be used 3 times per day.
A Flying Turtle has an acid breath attack that can be used three times per day and does 3-12 Hit Points damage.
The Fire Worm does 24-48 Hit Points of damage with its fire breath.
The Wurm Eater is a giant crab that can breathe out acid that does damage of 11-66 Hit Points.
Dragons and Hydras can have a breath weapon that is a 4d6 fire Killing Attack, which can badly burn a normal human being.
Clay and metal golems can breathe out a cone of fire four times per day.
Dragons of C'iel and Gaira in Anima: Beyond Fantasy. The former has a breath weapon that consists of feathers of light, that may instead cause the targets to stop the fight, and the latter one made of chains of darkness that end on spikes and can inmobilize their enemies. The game features also normal dragons, that have breath weapons a laDungeons & Dragons above.