I think it is better to not focus fire, but some players disagree and so we have the option. In most circumstances, especially if the battle is not a very even battle, you will see no difference whatsoever.
The difference is that with focus fire, your ships will try to stack up on AI ships as much as possible to kill each one as fast as possible. Without focus fire, aka the default, your ships will try to stack only a little bit, but to make sure they have basically "man to man coverage" on the AI ships, so that all the AI ships are taking damage as you go.
---
Why do man to man? Well, each time an enemy target is destroyed, your ships that were attacking it have to find a new target. They keep lists of potential targets, but in a big battle with a lot of ships moving around and blowing up, their list can get invalidated pretty fast. They're only able to go down that list so fast (same with the AI's ships), so if there's been a ton of explosions in the local area then they're going to waste possibly as much as a second or two before they find something else on their list that is still alive and that is the next most preferred target.
If it's not attack-move or FRD on those ships of yours, then you also need to add the qualifier "and is still in range." If they are in attack-move or FRD, then the first viable target that is still living is what they will go after -- even if that target has moved substantially. Ship target lists are updated on a rolling schedule, and there's a max number of target lists per second that can be updated (it also depends on how many targets there are, but it's usually hundreds or a thousand plus per second). So that means that if there's a really large battle, anyway, the targeting lists are being updated more slowly, and thus if ships are moving around substantially then each target-change is going to be that much less efficient.
In my view, doing man to man results in less downtime for your ships, and more bullets fired against enemy ships in the same amount of time. Thus, maximum overall damage output, especially in really huge battles with lots of movement. If the battle is small, not evenly matched, or not including ships moving around much, then the effect of this goes down.
---
Why do focus fire? Well, the theory here is that you kill off enemies faster, thus leading them to fire fewer bullets against you, thus leading you to take less damage and lose fewer ships, thus also allowing you to do more damage in return over the course of a long battle. I can see the appeal of this, and why some players feel like this is a better thing. In some cases it definitely is, and in some cases it definitely isn't.
---
So the big question is, should you be toggling this back and forth as you play? In my view, the answer is no. The AI doesn't use focus fire, and so you're not at a disadvantage by not using it. The game is balanced around not using it. If you use it, you might derive some advantage in very close battles without a lot of movement, but honestly most battles tend not to be that close. Either the AI is rolling you, or you're focusing your forces on its forces in such a way that you're hopefully rolling it in that one small engagement. The AI always globally outnumbers you 10000:1 or whatever, but I tend to try to make sure that I outnumber the AI 2:1 at most guard post engagements I am involved in, or to string out the AI on my planets so that I can engage them 2:1 or 3:1 or whatever as they stream along to me. And when you outnumber the AI, having focus fire OFF is definitely preferable.
In my view, which others disagree with, the only sensible time to use focus fire is if you're constantly in even battles that are fairly sedate in their movement.
