Guardian Forces (or GF) are the summoned creatures of Final Fantasy VIII. They allow each character to gain stat boosts, elemental/status offence and defense and commands which are used during battle by junctioning the GF. These summons also have a list of abilities and commands which they can learn through gaining AP points in battles while being junctioned. A GF can also have additional abilities learned by the use of certain items, as well as forgetting those they had previously learned.
Magic, within the world of Final Fantasy VIII, is utilized through Guardian Forces. These are, in many ways, similar to the summonings from other Final Fantasy games, with characters able to call them in battles, to carry out one strong attack.
However, a far more important facet of the Guardian Forces existence is the way in which they improve your characters statistics. Junctioning a GF to a character allows that character to use magic and other commands. It also, assuming the GF has unlocked the relevant ability, allows them to junction magic to a variety of stats, raising them to fantastical levels. Due to the GF system, it is easily possible to defeat Final Fantasy VIII without leveling up any characters, while attempting to complete the game without using GFs at all will be a much harder task.
GFs are collected in a variety of ways - by defeating them in battle, by drawing them from enemies, or by receiving them from other characters, either as part of the storyline or in a subquest.
In addition to having a major role in battles, the GFs are part of the main storyline as well. They are said to cause memory loss because GFs draw power from the same area of the mind that stores memories. Using their power does not come without consequences.
 GF Abilities
In order to have your GF learn an ability, first it must be junctioned to a battle active character. Then, go to the GF menu on the menu screen, and choose the GF you have junctioned. A list of it's abilities will appear; and the AP (ability points) required to learn them. Choose an ability that you wish to teach it. As the character it is junctioned to battles and gains exp, so will the GF gain exp, and AP. When the sufficient amount of AP has been collected, the GF will learn the ability. Note that abilities cannot be used until they have been fully learnt by the GF. Also note, that if you do not manually choose what abilities to teach your GF, it will automatically expend AP to learn it's abilities in a progressive manner.
There are also certain rare items you will collect throughout the game that can teach a GF any particular ability; and the item Gysahl Greens can be used to make a GF forget an ability, leaving room for another one to be learnt in it's place.
A GF has two types of abilities it can learn - Passive and Active.
A passive ability is one that goes into effect as soon as it is learnt; for example GF HP+40%, or Card Mod. Passive abilities do not need to be junctioned.
See Refine List for more information on the passive '-ref' ability.
The other type is active abilities - these are abilities that can be junctioned to your party. These themself can be split into two categories - Battle Commands and Support Abilities. Battle commands are abilities that can be used in battle; eg Card, or Draw. Support abilities are abilities that will augment the character with special skills or strengths - for example, a level up bonus or stat boost. Both must be junctioned in order to be in effect.
A full list of obtainable GF's and their abilities can be found below.