On occasion, you may get into a random battle that you immediately know is different.  The battle music is slower and gentler, and the creature will not attack you.  It will ask you for a gem.  If you give it what it asks for, you will gets lots of AP.  It will mention another friendly monster to you, and then disappear.  Each friendly monster will give you the gem desired by the next friendly monster, so you should always have what you need if you come across the creatures in order.
If you meet all eight of the first eight friendly monsters, you will be able to encounter the Friendly Yan on Vile Island.  The reward for giving it what it asks for is the ability to attack the optional boss, Ozma, with physical attacks.  This is a huge help if you intend to fight the game's hardest enemy.  Completing the Friendly Monster side quest can be dangerous, because some of them only inhabit dangerous areas...none more so than the Friendly Yan.  The regular Yans are very tough, but they are worth lots of AP and EXP.  If you run into them, use Odin and whatever other powerful skills and summons you have.  The Yans can use Snort and blow a character out of the battle, so be careful of them.

These are the Friendly Monsters in the order you should encounter them.  The locations given are merely where I found them - you may be able to meet many of them in other places, as well.