Flamer
"Build a man a fire, you keep him warm for a day. Set a man on fire, you keep him warm for the rest of his life..." -Terry Pratchett
The Flamer is a short-ranged support weapon employed in many different variations by nearly all of the races of the WH40K universe.
Overview
The flamer is a weapon that sprays an area with an extremely flammable liquid/gas known as Prometheum (no relation to the actual element Promethium but a reference to Greek God Titan Prometheus), burning anything in front of it. Now apart from the varying degrees of burns you'll receive when you're on the business end of one, your squad won't really be motivated when they see a giant firewave of painful and slow death, which will proceed to painfully and slowly kill them. In fact it's a rather terrifying experience, and anyone short of a space marine or a particularly stubborn commissar will probably bolt when they see a lot of 'em coming.
Armies field flamers in different sizes ranging from pistol-sized flame units, a regular rifle sized unit, to a big-ass flamer capable of bathing entire platoons in fire with a single burst. Table-top wise, Flamers are special weapons that ignore cover bonuses and are extremely effective against infantry. Very good against the Tyranids due to their total lack of vehicles, so you always have a chance at hurting most any tyranid unit you shoot at, plus it's great for killing all those pesky rippers. Fairly effective against Orks as well, as flamers are awfully good at killing large blobs of infantry. Still, they do fuck-all to most vehicles.
Imperial Variations
A cruel and painful weapon in its own right, this would not be 40k if there weren't several variations of the flamer, each deadlier than the last.
Hand Flamer
Mainly used by the Adepta Sororitas, Hive Gangers, and the Inquisition, this weapon is a pint-sized flamethrower that can be wielded as if it were a pistol. Some extremely unhinged individuals go as far to dual-wield them for extra burny power.
Flamer
The regular sized flamethrower as described above, used by nearly everyone in the 41st millennium.
Combi-Flamer
The Orks have proven that any weapon can be made more dakka by slapping another weapon onto it. Combining this with the Imperium's love for flame weapons, the combi-flamer is a weapon that can be strapped onto a regular Bolter to allow for a single gout of fire when needed.
Some may note that this is basically the weapon Ripley uses at the end of Aliens.
Heavy Flamer
The big brother of the regular flamer, able to do far more damage. Mainly used by Terminators and vehicles, they are also carried by the Sisters as a specialist weapon, by everyone in the Salamanders, and by the Catachans because they simply don't give a fuck. Strictly better than a regular flamer, which is why GW usually manually prevents you from just taking gobs of them. Unless you play a Crusade Legion list, where you can field squads of 10
Incinerator
The Grey Knights own personal flamers on steroids, dial up over 9000. Essentially exclusively wielded by the Grey Knights Purgation Squad (But can also be wielded by Terminators and Dreadnoughts as well). These fuckers have their fuel mixed in with blessed oils which when fired, essentially create a stream of flaming blue or white sanctic power. These guns cause most infantries to piss themselves red and scatter like droves, and make most regular Chaos forces to shit-launch themselves into near orbit of the sheer fear on the amount of collateral rape they will be forced to endure. Tl;dr, this is essentially the Grey Knights version of pesticide, to wipe out those hard to kill critters.
Flamestorm Cannon
An even bigger variation of the flamer, found on the Land Raider Redeemer as a sponson weapon and the primary weapon of the Baal Predator.
Inferno Cannon
The primary weapon of the Hellhound, this weapon focuses on the "thrower" aspect of "flamethrower". Similar in strength to the Heavy Flamer, the Inferno Cannon can project its payload over a considerable distance and if used right can fuck up even vehicles if they have poor armor.
Inferno Gun
Some men just want to watch the world burn. Others provide the matches. The Inferno Gun falls under the second category. Mounted only on the Warhound Scout Titan and the Malcador tank it will do horrifying damage against anything caught in its path.
Chaos Variations
Chaos Space Marines and Daemons both use flame weapons with fire generating from more warp-derived sources. Results of this are positively deadly as fuck! Additionally, CSM use a lot of the imperial flamer variants including the flamer, heavy flamer and combi-flamer.
Burning Brand of Skalathrax
Rumoured to be the weapon employed by Kharn himself to burn out his allies for being faggots and shying away from cold weather on Skalathrax, the Brand has a colourful history of burning shit to death - most notably power armoured Mary-Sues. This is a unique weapon that projects a gout of Khorne's (or Tzeentch? Nobody knows exactly) best fire at a long distance, to chase out and scorch anyone who might be hiding in cover.
Baleflamer
A larger, more common and much deadlier flamer that takes the Flamestorm Cannon one step further - in that it shoots further! These are exclusively seen on CSM Helldrakes and are super effective against almost anything NOT wearing terminator armour or hiding in METAL BAWKSES.
Breath of Chaos
Employed exclusively by Daemons, this is less of a weapon than it is a shooting attack - a shooting attack that eats through armour, body and spirit with ease, that is. Effects of this weapon are as varied as the Changer of Ways himself, but can include anything from simple conflagration up to fire that turns things into tentacles or frogs or - well, we can't really make an accurate assessment of this attack. Seen used by Flamers of Tzeentch mostly, but also by Daemon Princes and Lords of Change.
Bile Spewer
Technically not a flamer since it doesn't shoot fire but is still based on it. Papa Nurgle's forces doesn't use fire-based weapons because it doesn't leave anything for decay (since the flesh is burned off). Instead, his forces use "Bile Spewers", weapons using the flamer as the base weapon but the promethium tank is instead replaced with a tank full of toxic substances that does the same job as a flamer (efficiently kill infantry anywhere, be it in cover or in the open) but leaves a chance for the target's corpse to rot and decay, just as Papa Nurgle would like it. So far, this weapon's been only featured in Dawn of War II.
Xenos Variations
Though the Imperium has by far the most flame weapons, certain aliens also employ them.
Eldar Flamer
Used by Storm Guardians to support their role as close-quarters combat and by Wraithlords as secondary weapons for assault-oriented constructs.
Dragon Breath Flamer
Used by exarchs of the Fire Dragons, these weapons can sear anything that gets within range of their wielder. It only exists as a piece of fluff, as while there is an option to give one to a model on the tabletop, so far noone dumb enough to actually take that option over the basic fusion gun or fire lance played Eldar army.
Ork Flamer
The Orks employ Burnas for their flamer-duties. These weapons are unique because they can also be used as cutting torches to deal serious damage in close combat (AP3 marineopeners ftw! Unfortunately, they can only fire as a flamer or take on a better melee profile, but not both. FUCK YOU GW). They are also essentially a burning gas-can, strapped on a stick, to give you a hint on what they usually look like. Their vehicles occasionally employ larger versions known as Skorchas.
Tau Flamer
Used solely by Crisis Battlesuits, they are meant to deter close-range attackers. Due to the nature of said battlesuits, they are not intended to be used as a primary weapon (Farsight's pyromaniac buddy/possible fangirl Torchstar gives zero shits about this and she dual-wields them anyway).
Tau Plasma Flamer
Trust the Games Workshop to make a gatling version of any weapon, however inappropriate it is... Plasma Flamer is a triple-barreled gatling flamer, mounded on Y'Vahra battlesuit that shoots fire so hot it burns through terminator armour (one that is designed to work inside a nuclear reactor) and turns a marine inside into a crisp in a split second, and much like imperial flame cannons or chaos baleflamers it spurts that fire at a considerable range. Didn't I mention if fires in bursts? Because it does. Your army is now officially fucked.
Necron Gauntlet of Fire
Despite having ostensibly the most advanced weapons in the setting, the only Necron template weapon short of Apocalypse is the Lord-only Gauntlet of Fire. It projects a blast of green flame mechanically identical to a regular flamer, and can also reroll to hit and to wound in melee. So why don't you remember it? Because a melee-fighting Necron isn't going to not bring a Warscythe. You may see some more in 7th now that they can bring BOTH.