The battle system for Final Fantasy 4 is pretty much similar to the battle systems in previous Final Fantasies - meaning your characters are static and you attack in turn when your timer fills up. As always, you'll encounter monsters (excluding bosses) in random battles - the aspect of Final Fantasy which everyone loves to hate - and you'll have a choice of whether to FIGHT or to RUN.
It's a good idea to mostly choose to fight as you will level up quicker and therefore find the game a lot easier, however if the enemies in a certain area do not offer much in terms of Exp then it may not be worth the hassle. A disadvantage of running however is that you may occasionally lose Gil (how much depends on the strength of the enemy in question), also you may take a few hits before you are able to escape.
Weapons play a big part in the damage you can cause, as opposed to later games, where the character's personal strength attribute came more into play. Enemies can also be resistant to your weapons, or even absorb the damage to gain health. This depends on the element of the enemy and the element of your weapon.
However Final Fantasy 4 has a handy option if you notice this happening. If you choose ITEM from the battle menu and push UP, you can access the menu to change the weapons(s) your character has equipped. Try and make use of this option when you are battle enemies with varying elements to be able to dispatch them quickly.
You are able to determine how much damage a character takes from an attack (and in turn how much damage they can cause) by changing their position. This can only be done outside of battle via the FORM option in the main menu, and selecting a character in your party.
This will create a second pointer which you can use to move your chosen character up and down the list - the bottom spaces being the Back Row and the Front Row being the top spaces. It is recommended to have the mages in the back row and the fighter classes in the front.