I have a couple of characters who can use bows. I have also bought some arrows and have found some as well. My question is this: Do I have to use a quiver in order to fire the bow? Will the character continually shoot at available targets or will he kill one and then stop? Will he shoot all his arrows until out of ammo and then go melee? Thanks for the help!

Depends on how the AI is setup for that character. You can setup the AI to shoot at every available target. If you turn the AI off the character will not shoot.

Every character has a quiver on their inventory screen. You have to place ammo in these slots to use an equipped missile weapon. If you run out of ammo in the quiver, the game does not automatically re-load from your inventory -- you revert to whatever melee weapon you have equipped which, if you only carry a bow, means your fists. So your elven archer may end up in a fist fight with a lich if you don't watch your ammo carefully.