Living Weapon

Huntress: I take it that something went terribly wrong at that project--?
Dark Claw: Actually, the program was a resounding success. It produced exactly what it set out to produce: a remorseless killing machine devoid of all human compassion. A perfect weapon. They forgot one important thing about weapons. They are usually inert objects that can't get off the table and do harm by themselves. Their mistake was in creating a weapon with an imagination.

—Legends of the Dark Claw #1

This is the use of a living creature as a weapon. This is usually simply a subtrope of Organic Technology; an otherwise normal, user controlled weapon which happens to be a living thing. To qualify, it must be alive when it is used (so a slingshot made from a springy Y shaped branch doesn't count, but if it was still attached to a living tree it would) and it must be separate from the user (so claws don't count, since they're part of the creature that uses them; removing the claws from a creature and strapping them to your hand doesn't count since they're no longer a living thing, but a creature that happens to resemble a claw shaped glove that could be worn by another creature would).

This also has a few subtypes, examples of which go on their own pages;

Attack Animal - Trained creatures.
Equippable Ally - Other characters who are picked up and wielded as weapons (possibly after transforming into one).

Compare: Empathic Weapon, where the weapon has some sort of mind of its own but isn't necessarily "alive"; Improbable Weapon User, who could use one of these; Evolving Weapon, which can be justified by this; Living Ship for when the weapon is a ship; and Loyal Phlebotinum. See Natural Weapon for weapons which are part of the user but not independently living things.

Examples of Living Weapon include:

Anime and Manga

The EVA units from Neon Genesis Evangelion.
An interesting example is Chrona and Ragnarok from Soul Eater. The result of a witch's experiment, Crona had all of her blood replaced with a substance called "Black Blood" (not 'that' Black Blood) that was created in part by melting down the living weapon Ragnarok. The result is a mentally unstable Tyke Bomb whose weaponized blood has its own distinct personality.
In One Piece, the Kua all have Snake Weapons, large constrictor snakes that can change shape and stiffen themselves to be used as weapons (exactly how they do this and how they are trained is a mystery). Common uses for them are bows to shoot arrows, quarter staffs, stairs, chairs, ropes to tie up a prisoner, and so on.
The God Warrior from both the Nausicaä of the Valley of the Wind film and manga, able to slowly kill people around them with radiation, in addition to being able to fire a powerful laser beam from their mouth. The character is more developed in the manga than the film, getting a name (Ohma) and more time in the narrative. In the film, due to the shorter run time — and only a fraction of the manga being complete at the time of animation — they are only used once.

Film

Stitch was designed to be a bio-terror weapon.
This is what Weyland-Yutani wanted the xenomorphs for in the Alien films, and one of the common theories for their origins.

Literature

Every weapon of the Yuuzhan Vong, from the Star Wars Expanded Universe.
Harry Harrison's Eden books feature the Yilanè, a race of intelligent amphibious reptiles, who rely entirely on bio-technology, including Bio-weapons.
Shiftblades from The Deepgate Codex are souls of the dead conditioned to serve as Morph Weapons. Technically they're dead, but they have the characteristics of living weapons. Most remain in weapon shapes at all times, making them type 3, but one example in God of Clocks also takes human form and thus is type 2.
The two runeblades, Mournblade and Stormbringer, from The Elric Saga are soul-eating demons in the form of swords.
The Hork-Bajir in Animorphs probably fall into this...they are technically separate from the Yeerks controlling them, but are under their control as weapons.
The Speaking Gun from Simon R. Green's Nightside series. A conglemeration of flesh, bone, and gristle made to look like a weapon, it is a sentient creature that longs for the destruction of everything.

Tabletop Games

"Ophitech" in the roleplaying game The Day After Ragnarok falls under Type 3.
Magic: The Gathering has had occasional mentions such as the Dancing Scimitar and its spiritual sequel, Ensouled Scimitar. But there's also a whole group of artifacts called Living Weapons, which are equipment that come with their own hideous Phyrexian token creatures already attached. Considering that Phyrexian culture consists largely of mutilating and cyberneticizing each other into giant living weapons, and then beating the crap out of each other, this is pretty typical.

Video Games

World of Warcraft features a number of weapons that are alive. Such as the Dark Edge of Insanity, an axe that drops from an Old God. The axe itself appears to be covered with black reptilian skin, and is adorned by a huge yellow eye, that BLINKS and turns in its socket. Another example is Terestrian's Stranglestaff, the upper end of which consists of a rather disgusting octopus-like creature with squirming tentacles and a beak. And Zhar'doom, Greatstaff of the Devourer, is actually a Zergling/Felhunter-like creature stretched to the form of a staff. The head of the beast is the head of the staff, which clacks its jaws littered with very sharp teeth, and lashes forward with its tentacles.
Half Life has the Hivehand, basically a living alien beehive which shoots a constantly replenishing supply of bee-like aliens. There's also the Snarks. If you can get some from their nest, you can sic them on enemies, where they'll pester them rather effectively until they either get smashed or blow themselves up.
- The Opposing Force expansion also introduced the Shock Roach, the spore launcher (said to be Shock Trooper babies) and the Barnacle Grapple.
Geneforge has several type 3 weapons, most notably the "batons," gun stand-ins that eat and mate. (Also, creations are type 1, but they're Mons, so no matter.)
Most or all of the weapons in Prey.
The demonic sword Soul Edge appears to be alive and even has an enormous eye.
- Its opposite number Soul Calibur is one as well, albeit less prone to bodyjacking.
The Badgersaw from Postal 3 is kind of a cross of this type with type 1. It's a badger with a harness, so that you can hold it and have it maim your enemies.
A recent example, we have K9000 cyberdog gun from the Fallout: New Vegas DLC Old World Blues. It's a dog's brain stuck in a machinegun frame. It barks, whines, growls, and has two metal ear flaps and a little sensor/sniffer thingy that can detect enemies before you see them. Such a thing should really not be cute, but it manages, somehow.
In Metroid Prime: Hunters, Kanden's Volt Driver was stated in a Scan Log as being a Living Weapon.
The Tactics Ogre games have a spell/scroll called Snapdragon that allows you to turn an ally into one of these.
One of the calling cards of Oddworld: Stranger's Wrath was its spin on the idea of "live ammunition". To explain, Stranger uses tiny living creatures as ammo for his crossbow (each with various effects).
Daedric weapons and armor in The Elder Scrolls games are created by binding the spirits of lesser Daedra to the forging materials. In The Elder Scrolls V: Skyrim this is done by using Daedra hearts while smithing. Since Daedra are immortal (their physical bodies can be destroyed, but their spirits endure), this means that every Daedric weapon and armor piece is still alive. Fortunately, since each Daedra is an immortal Time Abyss, being used as a weapon or piece of armor for a century or so (Daedric artifacts vanish on their own after a while) isn't really a big deal.
In The Legend of Zelda: Skyward Sword Ghirahim, The Dragon, turns out to be one.
- The same game shows this is also true for the Master Sword, as it is Fi's true form. Most fans of the franchise will attest they like her much better this way, seeing as assuming her true form means she stops talking.
Kingdom Hearts has the Big Bad from the first game Ansem who fought with a monstrous Living Shadow known only as the Guardian, it seems to have no will of its own and merely acts as a weapon for the character.
In Super Princess Peach, the heroine's weapon and Non-Human Sidekick is Percy, a living, talking parasol.
Mr. Toots from Red Faction Armageddon, a baby unicorn you can find as part of the New Game+ for beating the game once, who farts incredibly destructive rainbows when you lift his tail. No, really.

Web Original

SCP-127, The Living Gun (warning, page image may be slightly Squicky!) from the SCP Foundation. On the outside, it's a totally normal MP5K. Except the magazine is locked in place. It shoots teeth - see the page image.

Western Animation

Willy McBean and His Magic Machine; King Arthur's sword Excalibur is alive and can talk, and seeing as Arthur has serious self-esteem issues in this story, provides a great deal of encouragement by doing so.

Real Life

Crabs and anemones.
- Large hermit crabs carry one or two on top of the shell. Predators generally go elsewhere when they see a brightly coloured bundle of venomous stingers waving to them - and those who don't quickly and painfully learn that they really should have. While the crab has thick carapace on top and isn't hurt if occasionally touched by a stinging tentacle.
- Small crabs cannot carry an anemone safely, especially when moulting. The Boxer Crab, however, grabs a pair in the pincers, which also allows to use them offensively. They look like cheerleader pom-poms. Needle studded, venomous (and the crabs wield the most toxic species commonly available in their habitat) pom-poms, with which the crab makes punching moves when it wants to discourage a critter. Of course, there are trade-offs. This being the boxer crabs' only defence, they are unwilling to drop these weapons (other than for moulting, and even then as briefly as possible), and their pincers have evolved toward having a better grip on an anemone without harming it... but crabs use pincers to eat. So not only they use anemones as "mops" to gather food from surfaces, but evolved the next pair of leg to cut and manipulate larger meals instead. Technically, this means the species in question learned to eat with both sporks and knives.