-=Galbadia Garden=-
There is no mention throughout the game of any SeeDs coming from Galbadia Garden, but the SeeD insignia is displayed at the entrance to the Garden. It is mentioned, however, that those that graduate from Galbadia Garden are often placed in the Galbadian Army.

-=Trabia Garden=-
In order to become a SeeD from Trabia Garden, you must transfer to Balamb Garden after taking your written test. At Balamb, you go through the field test, and into SeeDship.

-=Balamb Garden=-
To become a SeeD if you've studied at Balamb Garden, you have to pass the written test, the Fire Cavern field test, and then the field test.

So thats all well and good, but you want some more information on SeeD? Well, as I said, they are combat specialists, mercenaries that work for Balamb Garden. Each one is trained to handle just about any situation, from diplomacy to battle. The real reason behind SeeD was to train the 'legendary SeeD' that would defeat the Sorceress when the time came. Apparently this is Squall, even though any set of people in your party can defeat Ultimicia at the end of the game.

Now, if for some reason you didn't catch the symbolism between SeeD and Garden, let me explain, and explain why i'm explaining first off. I read somewhere someone asking one of the programmers of FFVIII how SeeD was pronounced, if it was Seed, or SeeDee. The reply was it could be either way, but I think it's just painfully obvious about the whole symbolism of SeeD. Edea looks at Squall while possessed and says 'A SeeD, planted in a rundown Garden'. Need I get blunter? There are three Gardens in the Final Fantasy VIII world: Balamb, Trabia, and Galbadia. Garden is a military acadamy that trains it's students to become combat specialists, and the elite make it into a group referred to as SeeDs. Those SeeDs are essentially raised in Garden, or grown in Garden. SeeDs come from Garden. Got it? You better by now...