In the relentless cheerfulness of the far future, the weapons used by the Tyranids are, just like their armies, home-grown. These weapons are organs, inseparable from their hosts and are counted as a single bio-organism. Some of them fire a "dumb" projectile, whereas other use seeds as their projectile, or even acids and living creatures. The downside to this is that bio-weapons are incapable of the explosive firepower that Imperial, Eldar and Tau guns have. They all trade this disadvantage in for being Assault weapons, meaning they can be fired on the move. Also, the explosive firepower is instead traded for more.... interesting and strategic effects. Another note is that due to their appearance, Tyranid weapons are some of the most badass looking in the entire of 40k.
Many of these weapons can be used by creatures of different sizes, but for the sake of simplicity the weapons have been ordered by the smallest creature that can use them. Small creatures are the equivalent of Gaunts, Medium are those comparable to Warriors and Biovores, and Large are those akin to Tyrants and Carnifexes. Then there are two other categories: the Inbuilt weapons, often part of the upper bodies of the largest organisms, and the Melee weapons, for the more choppy inclined creatures.
Small weapons are used by Termagaunts, the working class of any Tyranid army, and Gargoyles.
The Fleshborer is one of the most iconic Tyranid weapons, a simple design that fires a borer beetle that eats its way through anything it comes across during its flight. Being the basic weapon for Termagaunts, stats equal to the Bolter aside from half range and being an Assault weapon, this will prove to be more than enough to kill any unit of IG, Orks, other Gaunts or even basic Eldar units. Range 12, strength 4, AP 5, and it is an assault 1 weapon.
A lighter weapon, the Spinefists are used in pairs. They shower the target with a hail of poisoned spikes, the sheer amount of darts fired (as in: twin-linked) ensure that the target is hit. A tube runs from the air sac powering the weapon, through the user's arm to its lungs. This means that larger creatures get more mileage out of what is essentially the same weapon, proving that even something as simple as a blowgun can be horribly perverted by the grim darkness of the far future. Range 12", S3, AP5, Assault X, Twin-linked (where X is the number of attacks on the wielder's unmodified profile) The -only- time you want to take Spinefists is if you have to use up a few more points and want to give your Raveners or Rippers a ranged attack before they pile into close combat. All other models have better options for close to the same price.
 Spike Rifle
The Spike Rifle is a weapon exclusive to Termagaunts. It is, simply put, a harpoon launcher. That's about it. Statwise it's
about as far less imposing as than it's description. Range 18", S3, AP-, Assault 1. A great biomorph to take if you want to fake your opponent into thinking that you just started playing Warhammer 40,000 and don't know anything about anything.
With this weapon, your Tyranids too can learn how to shot web. The effects of the Strangleweb can be described best as a non-cutting variation of the Death Spinner used by the Warp Spiders: instead of slicing through their opponents it constricts them and maybe crushes them to death. That's a pretty big maybe: wounding even basic Guardsmen at 5+, and rolling against Strength instead of Toughness when trying to wound, this weapon would be best if used against the likes of the Death Guard ... if shooting them was a pretty stupid idea for a Tyranid player in the first place, AND if you could take more of them than 1 per 10 Gaunts.
The Devourer is simply put a lump of flesh that launches a shower of worms onto a target that immediately start to burrow into the target, crawl up to its brain and devour it. In addition to being fucking grimdark, this weapon has a nice range quite dakka for its size, albeit it doubles your Termagaunts in cost (while tripling their number of ranged attacks per round, a very economical way to increase dakka as long as you protect your gaunts somehow). If you take this army in large broods, you will be aiming at your enemy's biggest units to just drown the suckers in dozens of shots.
Medium weapons are used by Warriors and organisms based on the Vore and Guard bodies.
The traditional weapon of the Warrior, the Deathspitter works by stripping a large maggot of its shell and hurling it at the enemy. Its huge guts are highly corrosive and can splatter several enemies at once. Serving as a decent weapon against MEQs, you'll want to rely on your rate of fire to deal damage.
 Barbed Strangler
Remember that 70's cult classic Zardoz? Sean Connery dressed in red gimp getup and killing dudes like he's Khârn, at the behest of a giant floating head that claims that "The gun is good, the penis is evil." because the penis shoots seeds that create life? Well, the Barbed Strangler chooses to be both. The gun is very basic, a muscled tube that stores its seeds in a sack at the base, and shoots them with a powerful spasm. The seed upon impact greatly expands and sends tendrils to rip and tear through anything that gets in its way. Using the large blast template, you can cause a lot of damage to a blob in a single shot.
 Venom Cannon
Seen as the primary anti-tank weapon, the Venom Cannon launches a hail of corrosive crystals by way of a biological railgun. Though not as effective as you'd wish, the Venom Cannon can cause quite some hurt on Aspect Warriors and the like. Read: Ineffective against actual tanks, but okay against transports and open-topped vehicles. The Heavy Venom Cannon approaches actual strength values necessary to start glancing things like Land Raiders, if you roll really well, but there are still better options - and the only models with access to the HVC in the first place have other options available that are generally better (like two sets of twin linked devourers with brain leeches).
 Impaler Cannon
A recent addition, the Impaler Cannon is your answer to anything up to AV 12. This beautiful gun fires a bony spine, steered by a shard-beast as the base of the spine, whose innards are ripped from its body when the spike is fired. The spines are launched with enough force to rip through man and machine alike. Though its range is limited, you do not have to actually see your target to be able to shoot it, without a significant downside. Oh, and it has a higher rate of fire than the Imperium-equivalent, the Missile Launcher. What's not to like. This is your go-to weapon for dealing with METAL BAWKSES.
 Spore Mine Launcher
Created as a result of fights against the Imperial Guard, the Biovore displays this by being the Tyranids' closest thing to an artillery piece, and having a massive scrotum befitting a Guardsman's huge brazen balls. Armed with the Spore Mine Launcher, it can lob Spore Mines a considerable range. Though not as deadly as most other artillery weapons, the Spore Mines are unique in the way that if they don't land too close to the enemy, you can deploy them onto the battlefield as living mines, serving as an obstacle for your enemy that must be either moved around or shot. Similar to a real life dud. Generally more useful in larger point games, or Apocalypse.
 Flamespurt Cannon
It's a Heavy Flamer. Smarter than it's host, the Flamespurt Cannon serves as a bodyguard while the Pyrovore om nom noms the delicious biomass and make it ready for easy digestion. Instead of working together, the weapon and the beast function apart from each other, so both can dedicate themselves fully to their tasks. If they were used, anyway.
Here the biggest of weapons are described, used by the biggest of nasties.
 Brainleech Worms
The biggest of Tyranids have devourers loaded with Brainleech Worms instead, a species even more aggressive and hungry than the regular devourer worms. Range: 18", S6, AP-, Assault 6. If you're not using two sets of Twin-Linked Brainleech Devourers on your Flying Hive Tyrant and any Carnifexes you're fielding, you're doing it wrong.
 Stranglethorn Cannon
An upscaled version of the Barbed Strangler, the Stranglethorn Cannon has enough power to deal with even the toughest of foes, and serves as a borderline artillery weapon.
 Heavy Venom Cannon
The Heavy Venom Cannon is just that: a bigger version of the Venom Cannon able to deal with bigger vehicles, and it would be a lot more reliable if it wasn't using a Blast template.
 Acid Spray
Yo dawg I heard you like Hellhounds so I put a discharge of digestive fluids that works just like a Hellhound's weapon on your Tyranid MC and upped its cost by 100+ points. That's pretty much the gist of the Acid Spray. The problem is that you have a lot of weapons with stats like that, so using an expensive MC that attracts fire like there's no tomorrow.
 Fleshborer Hive
Think the Leman Russ Punisher, but replace the Punisher gun with a Fleshborer Hive. That's about it. Because sometimes you have just run out of Troops choices, and you just need 20 Termagants worth of Fleshborer fire.
 Rupture Cannon
If you want to stop a tank dead in its tracks, the Rupture Cannon is your friend. Firing two projectiles in a short succession, a tick that covers the enemy in a gooey substance, and a seed that upon impact with the goo dissolves at such speed it implodes with enough force to turn a Baneblade inside out. That's what the fluff says, anyway - it strikes at AP 4, so MEQs don't even notice it.
Certain weapons used by the Tyranids are part of the creatures themselves, instead of being addition to the arms/back.
 Flesh Hooks
Until the current edition came out, nearly any species of Gaunt, Warrior and more could use these. They are long bony spines with barbed hooks on the end that serve as both harpoons and possible grapnel for climbing. Now they're Lictor only, count as Frag Grenades, and can be fired for a surprisingly deadly shot.
Exclusive to Carnifexes, it is a biological equivalent of a shot from a plasma cannon, and just as deadly. However, since Monstrous Creatures can only fire 2 of their ranged weapons per Shooting Phase, and you should already have two sets of twin-linked brainleech devourers on your 'fex, you'll never get the chance to use this biomorph and shouldn't waste the points to begin with.
 Bio-electric Pulse
Built into the Trygon, a electrical charge is build up by the creature as it moves, allowing a powerful (and large) discharge. The Trygon Prime increases the range of this blast, and doubles its rate of fire.
In action: https://www.youtube.com/watch?feature=player_detailpage&v=mtY3M-mwdZY#t=13s
 Ripper Tentacles
Used by Mycetic Spores, these are long-ranged tentacles able to do significant damage to medium-sized blobs thanks to their high rate of fire (strange choice of words when the weapon in question is akin to beating someone over the head with a tentacle)
 Spore Mine Cysts
Build into the underside of a Harpy, you can have it drop a Spore Mine onto something it flew over once per game as if it was making a bombing run. The same rules for the Spore Mines as stated above count here as well.
 Cluster Spines
The largest Tyranids have banks of spines build into their carapaces that they can fire when under stress. These hollow spines explode upon impact, dealing significant damage to anything in the area of impact.
 Stinger Salvo
Works comparable to the Cluster Spines, except this is more of a barrage of large solid spikes akin to the spinefists.
 Thorax Swarm
Certain Tyranids can issue forth swarms of bugs living in their thoraxes to devour anything in short range. These swarms come in three flavors: Electroshock Grubs who electrocute anything they grab onto for a damaging attack, the Desiccator Larvae who drain the target's body of all of its moisture, and the Shreddershard Beetles who Rend their way through their targets, exploding in shards of needles.
Every Tyranid organism has at least rows of sharp teeth and deadly claws, but some go an extra mile.
 Scything Talons
The long, bladed weapons you see on Tyranids of all sizes, they allow a fast barrage of blows onto a target. This is shown in-game as rerolls: the more you have the more likely you are to hit your enemy.
 Rending Claws
The iconic claws of Genestealers and the origin of the Rending rule, these claws are both incredibly sharp and hard, allowing you to pick apart even a Terminator with ease.
Psychic swords that feed off the life force of their target, they can inflict Instant Death, and are more likely to do so if used in tandem.
 Lash Whip
A classic and the source of many a rule 34, these serve to slow down a target and force it to strike at Initiative 1, allowing the user to strike first, and often to kill. This sounds great until you read the Tyranid FAQ that says that Lash Whips apply at the beginning of the Fight sub-phase, and then other modifiers come into play. Meaning that Banshee squad you just walked your Hive Tyrant into is still striking at Initiative 10 because of Banshee Masks. Make sure you know exactly what you're dealing with before you attempt shenanigans with Lash Whips, as the only models with access to them are going to be expensive (Warriors, Shrikes, Hive Tyrant/Tyrant Guard).
 Crushing Claws
Used only by Carnifexes and Tervigons this massive pair of crab-like claws grants d3 extra attacks to the model but forces it to strike at Initiative 1, regardless of whether your Carnifex charged that turn (Living Battering Ram is overridden by the claws penalty, read the FAQ). An okay option for the Tervigon, if you actually plan for it to get into close combat, but its role is really more sitting on an objective, shitting out as many gaunts as it can, and then buffing those gaunts with either its default powers or powers rolled on Biomancy or Telepathy. Carnifexes on the other hand have pretty much lost their utility as linebreakers, you're better off running them as a dakkafex and letting Trygons do the heavy lifting.