Unlike the summons in FFVII you can't just pick up these GFs most of the time you will have to defeat them in battle to gain there respect. When you get to the more powerful summons near the end of the game this can be a hard task. There are some GFs that you get in different ways. Shiva and Quetzalcoti you already have when you start the game, others like Siren you can draw from bosses. When you aquire a new GF you can then call on their powers in battle. The GFs the same as your characters gain levels so that they will get more powerful as you go though the game. This prevents them from become to weak to bother with by the end of the game which happened in FFVII with the earlier summons.
In FFVIII the GFs are your main source of abilities if you have no GFs junctioned then the only command you will have is attack. When you junction a new GF a whole range of abilities are made accessible to the character junctioned. If you have played FFVII then in order to get to grips with the juntion system you just have to think of the abilities as materia. This is because the same as materia each ability has to be junctioned to the character individually. When you summon a GF in battle your characters time bar will change from yellow to blue which is then the count down to your GF being cast. While the bar is blue the GF will take any hits directed at your character. This might sound good but be warned your GFs only have a certain amount of life too and once their killed you can't revive them in battle. Out of battle they can only be revived by G-Returner which you can buy from pet shops. As there levels go up they will become more and more powerfull. I suggest that the first ability you get your GF to learn is Boost as this can double the strength of your GF most GFs have this ability but not all of them. To use boost you just have hold select during the GF introduction sequance. A white glove should appear in the bottom corner of the screen basically you have to tap the square button as fast as you can but stop when the red cross covers it. If you tap while the red cross is there the number will fall back to the starting value. If you reach 150 thenthe strengh of your GF should be doubled.
Not all GFs can be junctioned ones like Odin and Gilgamesh just appear at random during battles, but when they do appear they deliver very powerful attacks which have got me out of some very sticky situations.