Twin-Linked weapons are a subtype of weapon found in the Warhammer 40,000 universe that embody the timeless tactic in warfare of ensuring a hit by slinging more rounds at the target. The way this is done is typically duct taping two guns together and firing them at once so that the target has to avoid twice the number of rounds. Typically vehicles, monstrous creatures and terminators carry twin-linked weapons, considering that typically only heavy weapons are twin-linked. Although storm bolters are twin-linked boltguns in appearance, they actually increase the fire rate of the weapon. Perhaps a prerequisite of being allowed to wear Terminator armor is knowing where those extra shots go.
Where exactly the other gun's shots go if you hit the first time is a mystery whose answers are known to none, but it means that twin-linked weapons are, somewhat ironically, horribly inaccurate as at least one of the guns is always guaranteed to miss the enemy. As of 8th Edition, this is now no longer an issue, see below.
In Game Terms
When trying to hit a target, you can reroll if you miss. This is useful especially if you are using a weapon that has a good chance of melting your face off rather than firing, as it means that if you get a Gets Hot! result you can reroll, though template Gets Hot! weapons roll for it before resolving scatter, so don't put too much faith in plasma cannons. Weapons that scatter reroll an undesirable result if they are twin-linked as well. This can be exploited to allow you to reroll ordnance (for example, coupling a Triarch Stalker with a Doomsday Ark).
This, however, resulted in a rather odd display of mechanics over common sense: if you hit with your first shot, rolling the second bullet would cause it to just disappear into the warp itself. This made twinlinking always flatly worse then just having a second shot: Two of the same gun have the same odds to get at least one shot, but the two guns have a chance to score a hit and wound with the second shot.
Twin-linked no longer gives you a reroll - in an incredible display of common sense, our overlords have decided that a twin-linked autocannon is simply two autocannons duct taped together, and twin-linked weapons now give you twice as many shots as a non twin-linked weapon.
Aside from the logical improvement to its advantages, since the additional shots now actually go somewhere (they can all hit the target), it also logically makes twin-linked plasma weapons more likely to melt your own face, instead of less likely. Which really does make sense.