A forcing cone is a part of a firearm, usually either a revolver or a shotgun. It plays a slightly different role in each type of gun. In a revolver, the cartridge is kept entirely in the cylinder during firing. The front of the bullet must, of necessity, be "behind' the rear end of the barrel since the cylinder must be able to rotate the bullet into firing position behind the barrel. As a result, when the bullet is fired, it is not physically within the barrel yet. Since there is always a chance (due to shock, or wear, or simply perhaps normal variation of parts) that the bullet is not perfectly aligned with the barrel, there is a small cone which protrudes from the rear of the barrel, wide end towards the cylinder. The rear (wide) end is slightly larger than the barrel, and the front (within the barrel) should be the same diameter as the interior of the lands (if it's a rifled barrel). This will 'force' the bullet into the proper alignment to travel down the barrel. If this forcing cone is damaged or misaligned, or if the cylinder comes to rest too far out of alignment, the revolver may 'spit.' This means to release burning gunpowder and perhaps even slivers of bullet material sideways when fired, as the misaligned bullet strikes the edge of the cone and shaves off fragments and the powder on the 'blocked' side is diverted sideways. This can pose a hazard to the shooter or bystanders. Bullets thus damaged will often also suffer reduced accuracy due to the deformation, and the impact on the cone will increase wear on the cone and frame dramatically. As an aside, this may mean that it is much safer and more accurate to shoot ball ammunition in a worn revolver rather than wadcutters, as the rounded head of a ball round will 'seat' itself more gently.
In a shotgun, the forcing cone is found in much the same place - at the breech end of the barrel - but performs a slightly different task. Shotgun shells are slightly wider than the bore of the gun, and they are also made of material that can compress around the edges, rendering their openings slightly wider at firing. The forcing cone in a shotgun is intended to compensate for this, and to direct the mass of wadding and shot down the barrel smoothly. In shotguns, the longer the forcing cone, the better the shot pattern (generally). Older, more expensive shotguns sometimes offered barrels which tapered all the way from the breech to the choke, which produced the least disruption of the shot and pattern. Very short forcing cones can lead to deformed shot as the mass of shot are forced through a rapid compression without time to align themselves.