Friendly Monsters are a group of enemies that can be encountered on the world map, upon which entering battle with them, will ask for an item from the player. Should you give them that item, then they will give you an item, as well as a bunch of AP. However, the Friendly Monsters will run away if you attack them (with the exception of the final one), although they can be encountered again. You will know that you've found a Friendly Monster, as different battle music will play, which is the same music you hear when you find the Ragtime Mouse.
Although they are random encounters, one can do the stutter step technique (just tap the D-Pad) on the world map to reduce the chances of finding normal enemies, Friendly Monsters can still be found using it. Also, since they give a large amount of AP, it might be best to wait until you need that AP to master an expensive ability. The Friendly Monsters do not have to be found in the order listed on this page, although that will be the natural progression of them and you should probably grab them while going through the main story. The only thing that matters is that you found all of the others before going after Yan, as he is the final one and will run away if you haven't found them all.
Mu is likely the first Friendly Monster you will come across and can be found in the small area of the world map just outside of the Village of Dali. He wants a single Ore and will reward you with a Potion and 10 AP.
Ghost can be found on the plains outside of the city of Treno. He requests a single Ore and for your reward, he will cough up a Hi-Potion and 10 AP.
Ladybug is found in the forests around both Conde Petie and Black Mage Village. The large forest surrounding Black Mage Village is an ideal spot, as there is plenty of room to hunt her down. Ladybug wants 2 Ores and in exchange for that, she will hand over an Ether and 20 AP.
After finishing up the events at Conde Petie Mountain Path, you will be in the area with Madain Sari. Head into one of the forests and keep fighting until you find Yeti. Like with Ladybug, he'll ask for 2 Ores, but will reward you with an Elixir and 20 AP.
You will find Nymph in the area where the Iifa Tree is located on the Outer Continent. Once you pass through the Conde Petie Mountain Path, there will be a forest right next to you. Head here and keep fighting until you encounter the Nymph, who will ask for 3 Ores. Your reward for doing this will be an Emerald and 30 AP.
Jabberwock can be found in the forests on the Forgotten Continent, with your first opportunity to find him being during the events of Disc 3, as this place is inaccessible without an airship and you are forced onto the continent during the story. Once you find him in the forests, he will ask for an Emerald and will reward you with a Moonstone and 40 AP.
On the frozen Lost Continent, which is the landmass in the northwestern part of the world map, you will eventually find this Friendly Monster. Note that this one can be an elusive one, but if you concentrate on the chocobo tracks just north of the shore near Esto Gaza, you will have better luck (like, actually running along the tracks). It will ask for a Moonstone, which doesn't affect that trophy, and will reward you with 30 AP and a Lapis Lazuli.
If you've ever tried to track down a Grand Dragon in the area above Gizamaluke's Grotto, then you should know where Garuda's location is. If not, there is a vine in the one room in the back of the Grotto, with the two moogles, where you can climb up and reach an area known as Popos Heights. This can be a very dangerous place, because of said Grand Dragons, but stick to the forest where you emerge to hunt down this Friendly Monster. It will ask for a Lapis Lazuli and reward you with a Diamond and 40 AP.
As the final Friendly Monster, you will need to have found all others and given what they asked for, since Yan will run away if you didn't. The friendly Yan will be found on Vile Island, which is the island just southwest of the Iifa Tree; it is only accessible with a flying Chocobo or with the Invincible airship (the Hilda Garde III doesn't have a spot to land on the island). Should you have found all the others, then the Yan will ask for a Diamond and will reward you with both a Rosetta Ring and 50 AP.
Note that the friendly Yan is the only Friendly Monster that will not run away should you attack it. Before you do, though, it should be known that it is definitely one of the most dangerous battles in the game. Defeating the friendly Yan will reward you with the same item and AP as above, plus you can find Yan over and over, as long as you don't have over a Diamond. His only move is a physical attack that hits all characters and has a chance to inflict Silence, but it definitely hurts and Yan will counter everything with it, too. The creature has high defenses, so make sure you use defense-ignoring skills, like Thievery, Dragon's Crest, Frog Drop and/or Limit Glove.