I assume the red bar is fatigue, but what effect does it actually have on gameplay? It doesn't seem to go down much...

If you run out of it you collapse. Only resting restores it. So it's basically how soon you need to sleep.

Daggerfall used this kind of fatigue too. They changed it to the short-term kind of fatigue in Morrowind.

In Daggerfall it definitely had an apparent effect, though. I never once noticed it in Arena.

Maybe it depends on your character, but for me it tended to drop pretty fast in long fights with enemies and especially when swimming. I found myself relying on my Bracers of Stamina quite a bit during longer dungeon crawls.

If you get to zero, you die. It drains most quickly when you're swimming, but even just walking around drains it a bit.

I do not believe it has any effect on fighting ability.

O.K., I'm going to expand a bit on my hasty answer.

In Arena, when your fatigue gets to zero, you fall asleep. There are three possible outcomes of that sleep. If you are swimming, you drown. If you are on dry ground, and there are no enemies nearby, you sleep and wake up slightly refreshed. If you are on dry ground, and if there are enemies nearby, they kill you.

As a practical matter, fatigue isn't a factor in Arena. Fatigue barely drains except when you're swimming, and even then it's not so bad. So unless you're making a point of healing only via magic, you will have to sleep due a lack of to hit points or a lack of spell points before you have to sleep due to fatigue. The only class I can see fatigue being a factor for is Sorcerer.