'Linked' means that, when you have that weapon group selected for active control, a mouse-click will cause every weapon to fire at once.
'Alternating' means that each click will fire one weapon, then swap to the next - it's primarily useful for weapon groups with missile racks.
Alternating can also be useful to get a steady (or steadier) stream of fire out of multiple slow-firing weapons; for example, a Dominator with a pair of gauss cannons - if you fire both at once, the enemy can tank that on shields, then drop shields and dissipate hard flux while the guns cycle, where if you fire one at a time, that doesn't work as well (or potentially at all).
There are, I think, some additional subtleties regarding whether or not weapons fire based on firing arcs? I don't recall the details offhand.
And weapon groups that are set to autofire will completely ignore the setting; when on autofire, each individual gun gets its own little targeting AI that chooses whether or not to fire separately from every other weapon in the group.