A specific type of Artificial Limbs that's Exactly What It Says on the Tin: a ranged weapon integrated with a character's arm. Or their leg; we're pretty flexible here, although they'd have to be good at shooting while crouched, as you'd probably have to be on all fours or sitting to aim a leg-mounted weapon. And it doesn't have to be literally a cannon; a machine gun or multi-shot rifle or shotgun is also acceptable. Generally, his trope includes any ranged weapon or gun used in place of (or as) a prosthetic.
While impressive these weapons are far from practical for numerous reasons. First, the recoil from an actual cannon would tear the the rest of the limb off, potentially killing the operator. Second, any ranged weapon (machine gun, shotgun, rifle, handgun) will have some recoil; if it's attached to the person (as a prosthetic would be) the recoil would at least mess up their aim for additional shots, and probably spin them around. Third, any weapon large enough to to qualify as a cannon would be so much weight on one limb that it would be next to impossible to maintain balance or get a steady enough stance to aim it, much less use the arm in melee combat. Fourth, barring some sort of advanced cooling system, any gun like weapon will heat when fired, almost certainly injuring a user keeping the gun directly on their skin. Five, the placement of the gun would make loading ammo or maintenance awkward at best and severely limit the amount of ammo that could be carried. Sixth, triggering an arm weapon requires either mechanical design that may require the other arm or exotic technology to connect directly to the brain. Seventh, having a loaded gun on or replacing a limb is just begging for accidental discharges when doing mundane tasks, even just walking with it.
There are two general types. One is the complete replacement of a hand, or even an entire arm, with a gun. Realistically, these make very little sense, as getting an Artificial Limb would allow you to hold a gun, and do other mundane tasks with the limb when you're not in combat. Arm Cannons that fully replace the limb are also prone to suffering Fake Arm Disarms, as they can be destroyed without gravely injuring the wielder. A specialized version is a robotic hand that turns into a gun. This version often overlaps with the Hand Blast trope.
The other, more mundane version, is simply a gun that's attached to the arm — generally with the hand still intact. This might also involve Artificial Limbs, but just as often uses a special harness or weaponised vambrace. Occasionally, the weapon may fit over the forearm and hand, giving the appearance of the first kind. More discreet versions often overlap with Nothing Up My Sleeve. Aiming it is usually a simple matter of pointing in the general direction of whatever it is you want to shoot. Using your free arm to brace the cannon before firing is optional, partly because reducing weapon sway and/or recoil improves your accuracy, but mostly because it makes the resulting shot look more powerful (or at least more awesome).
Both kinds generally come with Bottomless Magazines, and are often fired by thought alone. Larger versions of both types are common on Humongous Mecha, which can be conveniently designed to carry them from the start.
At any case, some arm cannons look like trumpets by design.
For a weaponised arm that isn't a gun, see Power Fist, Rocket Punch, and especially Blade Below the Shoulder (and sometimes Pile Bunker). Not to be confused with Hand Cannon. Or, for that matter, Armed with Canon. Compare Shoulder Cannon, Leg Cannon, and Hand Blast. Finger Firearms is a small subtrope.
