The battle system in Final Fantasy 8 is centered around the junctioning system. Although easy to learn and get the hang of, junctioning is very deep and complex.
You start out by junctioning a GF to a character. If you don't assign a GF to a character he/she can't do anything but regular attacks.
After junctioning a GF, you can pick 4 command abilities that you want that character to have. The main five are attack, magic, draw, GF, and item, but you will learn many more as you progress through the game. You can then assign 3-5 (it can be more than 3 if you get certain items) bonus abilities, like HP+40, Auto Haste, and so on. These are sort of like the accessories of Final Fantasy 8, they allow you to enhance a character's stats and what they do in battle.
Learning New Abilities
Once you junction a GF, you can then decide what abilities you want him/her to learn. The abilities, once learned, can be assigned to that character. To choose what you want a GF to learn, go to junction, then go to GF, and select the one you want to see. A menu will pop up showing what they have already learned and what they can learn. A number to the right of the abilities not yet learned will show how much more AP will be needed for the GF to learn that ability. One thing to be careful of is to not use the magic that you are junctioning too often. For example, if you junction cure to your strength, and then constantly use the Cure magic in battle, your strength will get weaker and weaker every time. You can tell if a magic is junctioned to you in battle by a little symbol next to it. Also keep in mind that the maximum HP you can get is 9999, and the maximum of everything else is 255. See the magic section to find out how much each magic improves a character's status when junctioning it.
What Can Be Junctioned
Another important part of the junction system is junctioning magic to a character. The tricky part is, you can only junction magic to the statistics the GF has already learned. For example, you can't junction magic to a character's HP until one of the GFs junctioned to that character has learned HP-J (HP Junction). You can then go to magic on the junction screen, select HP, and then choose a magic to junction to it. Once you highlight a magic, it will show how much that magic improves (or worsens) that statistic. Usually, the more powerful the magic, the better it junctions to a statistic. Another important factor is how much of that magic you have. If you have the maximum (100), it will improve that statistic much more than if you had 50 or so. Always try to get 100 of every magic whenever possible.
After a GF learns Elem-Atk-J, the character junctioned to that GF can junctioned elemental magic to their weapon. For example, if Squall junctions 100 Thundagas to his Elem-Atk-J, his attacks in battle will be heavily Thunder based. Use Elem-Atk-J to junction magic that is the weakness of your enemies. For example, in the Fire Cavern, you should junction Ice magic to your weapon, and so on. Keep in mind that only elemental magic can be junctioned here. Just like regular magic, the more of that spell you have, the more it protects you.
Elem-Def-J is just like Elem-Atk-J, except instead of putting it on your sword it makes it your defense. This can only be used once a GF junctioned to that character learns it. For example, in the Fire Cavern, you should junction 100 Fires to your Elem-Def-J to be completely immune to the character's fire attacks. If you get into the more high level magic, you can even absorb enemy's elemental attacks. Just as before, the more of that kind of magic you have, the more effective it is in defending you. Another thing to keep in mind is that some GFs can learn Elem-Defx2, 3, or 4. This allows you to junction 2-4 elemental magics to your armor instead of just one.
ST-Atk-J and ST-Atk-D
Another way you can junction magic is by junctioning it to your status attack and defense. Once a GF learns ST-Atk-J or ST-Atk-D, you can junction status based magic onto your weapon or armor. For example, junction 100 Sleeps to your weapon and you will put an enemy to sleep every single time after you attack them (unless they are immune to it). And if you junction 100 Sleeps to your defense, you will never have to worry about the Sleep status (except on some very rare occasions from a special boss or enemy). Keep in mind that only Status magic can be junctioned here, and the amount of the spell you junction will represent the percent. For example, 80 Zombies in your weapon will make you inflict the Zombie status 80% of the time, and 80 Zombies in your armor will protect you from the Zombie status 80% of the time. Just like Elem-Defx2, 3, or 4, you can have ST-Defx2, 3, or 4, which allows you to junction 2, 3, or 4 status magics to your defense.
That's about all you need to know about the junction system. With a little practice you'll be junctioning like a pro in no time.