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.
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 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.
GURPS has the dragon template, which of course includes a Breath Weapon
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 Thirteenth 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.