This article is from the Final-Fantasy and other Square Soft Games FAQ, by email@example.com (Nick Zitzmann) with numerous contributions by others.
Here are the GFs which you can collect/junction/summon by normal means:
Quetzalcoatl and Shiva (Disc 1): You start the game with these two.
Ifrit (Disc 1): Acquired in the Fire Cave.
Siren* (Discs 1 & 4): Can be drawn from Elvoret during the mission in
Diablos (All Discs): Acquired by using the Magic Lamp item. You get
the Magic Lamp by talking to Cid following the
dance scene. (It can't be acquired at any later
time in the game.)
Brothers (Discs 1 & 4): Acquired in the Tomb of the Unknown King.
Carbuncle* (Discs 1 & 4): Can be drawn from Iguion during the mission
in Deling City.
Leviathan* (Discs 2 & 4): Can be drawn from NORG during the revisit to
Pandemona* (Discs 2 & 4): Can be drawn from Fujin during the mission in
Cerebus* (Discs 2 & 4): Acquired in Galbadia Garden.
Alexander* (Discs 2 & 4): Can be drawn from Edea in Galbadia Garden.
Doomtrain (Discs 3 & 4): Acquired by using the Solomon's Ring item, 6
Steel Pipes, 6 Remedy+ (requires Alexander for refining
through Med LV-Up), and 6 Malboro Tentacles. Note that
Solomon's Ring can only be found at Tears' Point, and only
before Lunatic Pandora arrives at the point (in other words,
it's only available after entering Esthar and before
Bahamut (Discs 3 & 4): Acquired at an underwater base. It's in the
southwesternmost corner of the map, and requires approaching
a blue light. Pick the first option first, the second option
second, and the third (invisible) option third.
Tonberry (Discs 2, 3, 4): Acquired in the Centra Ruins. First, acquire
the GF Odin. Then, fight 20 Tonberries. After defeating all of
them, you'll fight the King Tonberry immediately afterward.
Cactaur (Discs 3 & 4): Acquired on Cactus Island, in Centra. (You must
defeat the Giant Cactaur to receive this GF.)
Eden* (Discs 3 & 4): Can be drawn from Ultima Weapon in the underwater
base. To get there, you must have Zell in your party, and you
must spend all but ten units of power getting to the bottom of
the base. At the bottom, let Zell handle opening the door.
Here are the GFs which can only be summoned, but not junctioned:
Odin (Discs 2, 3, 4): Enter the Centra Ruins and complete it within
the time limit. Get both eyes from the gargoyle statues, and
place them both in the statue on top to receive a passcode.
Take the eyes, place them in the gargoyle down below, and then
enter the passcode to fight Odin. (Odin will randomly enter a
battle and destroy all the enemies, but the chance of him
appearing is random and depends on luck.)
Gilgamesh (Disc 3): If you got Odin before the end of disc 3, then
Odin will become Gilgamesh.
Phoenix (Discs 2, 3, 4): Summoned by using a Phoenix Pinion, which
can be refined from Mega Phoenixes (which can be refined
from Phoenix Downs).
Chicobo (Discs 2, 3, 4): Summoned by using Gyashl Greens, which can
be purchased from Chocoboy inside Chocobo Forests.
MiniMog, Moomba (Discs 2, 3, 4): These two are only available through
the PocketStation "Chocobo World" minigame.
* = If you missed drawing out this GF at this point in time, and if
you're playing the non-Japanese version of the game, you can receive it
again inside the final dungeon during disc 4.
Also, note that for the junctionable GFs which are not drawn out of
enemies, you will need to fight and defeat the GFs to earn their