What does constitution do, exactly? Is it just how long it takes to get fatigued, or does it affect HP-gains etc as well?

Of course Constitution affects your hp. You should always try, your main char to have 18 con. Especially if youre a fighter you get more hp bonus if your constitution is more than 17+. I think that if you have a really bad constitution like aerie, you get faster tired too.

Only fighter classes benefit from a 17 or 18 Constitution. Unless you have a fighter class, 16 gives maximum benefit.

In paper D&D, Constitution also affects your ability to be resurrected.

Yes, how'd it go? You could only be resurrected an amount of times equal to your natural constitution, which also determined the percentage chance of being obliterated instead of raised. Let's not forget the system shock survival tallies, either. Constitution doesn't seem to play an incredibly important role in BG.