To Flee is one of the primary commands of battle in the Golden Sun series. Like many other RPGs, at the start of each turn you may select the Flee button so that you may attempt to end the battle right away, without any EXP or coin rewards. Also, perhaps as expected, the chance to flee is lower if your enemies in battle are stronger and/or have a higher Agility rating than you. You may try to flee and get the message "but there's no escape!" This will take up your entire turn; that is, if you fail, your characters will do nothing for a turn while your opponent attacks. And of course, you cannot flee from any scripted battles relevant to the plot. If you try to flee from these sorts of battles, you will simply receive a message that says "This is no time to be running away!" and you will be returned to the primary commands.
There are several monsters that have the ability to flee from battles as well, which is distinctly different from them being felled, as any monster that has fled the battle will not drop items or coins, nor reward EXP. Furthermore, the Jupiter Djinni, Gale has a chance to "blow an enemy far away", which is identical to forcing the enemy to flee (and hence does not reward the player for their efforts).
Perhaps the only way Golden Sun games handle Fleeing from battles differently to any other RPG is that, if your "party attacks first" in a normal battle (the common "Pre-emptive Strike" also featured in many RPGs), if you choose to Flee on your one free turn of the battle, you will be guaranteed to Flee successfully regardless of your opponent's statistics.
Obviously, a player should not use this command every single battle, as it will result in the player failing to gain levels, which will make the game progressively harder. Thus, the most appropriate time to use the Flee command is when the party is nearly downed or to save time when crossing stretches of the map that contain weaker enemies.