Rare Cards
Rare Cards are great to help you win more games as they have very good stats.  Also, they can be refined into many useful items.  To see what you can refine from those cards, click here.
Alexander
Win from Balamb Hotel owner.

Angelo
Win from Rinoa's pals in the train during your first mission.

Bahamut
Received after battle with Bahamut.  For more information about getting this GF, click
here.

Carbuncle
Win from Hearts in the CC Group Quest.  For more information about this side quest, click
here.

Diablos
Received after battle with Diablos.  For more information about getting this GF, click
here.

Edea
Edea has her own card.  You can win it from her when she is at the Centra Orphanage.

Eden
Received after battle with Ultima Weapon.  For more information about getting this GF, click
here.

Gilgamesh
Win from the King in CC Group Quest.  For more information about this side quest, click
here.

Ifrit
Received after the battle with Ifrit in the Fire Cavern.  For more information about getting this GF, click
here.

Laguna
Win from Ellone in the Space Station.

Leviathan
Win from the Joker in the CC Group Quest.  For more information about this side quest, click
here.

Minotaur
Received after battle with Brothers in the Tomb of the Unkown King.  For more information about getting this GF, click
here.

Quezacotl
Win from the mayor of Fisherman's Horizon (or his wife... must check up on this).

Quistis
Go to the Balamb Garden Canteen.  Play the guy sitting on the right who is a Trepie.  He has the Quistis card.

Rinoa
Play Caraway.  Give him your Ifrit card and he will play the Rinoa card.  After getting the Rinoa card, win the Ifrit card from Headmaster Martine in Fisherman's Horizon (he is to the left of the mayor).

Sacred
Received after battle with Brothers in the Tomb of the Unkown King.  For more information about getting this GF, click
here.

Seifer
Win from Headmaster Cid

Selphie
Win from her friend near the gargoyle statue in Trabia Garden.  You can play her after you leave Trabia Garden.  Re-enter the Garden and play her.

Siren
Go to the pub in Dollet and play the guy upstairs who will tell you "So are you ready to start?"  Answer Yes all the time, and then you will play cards.  When you win him, he will take you to his secret room.  Make sure you snoop around his magazines, there are some items in them.  Then talk to him and he will give you lots of low level cards.  Since in Dollet you play by the Random rule, this is a big disadvantage, so before asking him to play, refine the low level cards he gave you using the Card-Mod ability learnt by GF Quezacotl.  Keep one copy of each card for yourself, however.  Play him and he will now use the Siren card.  If he doesn't play it in the first hand, keep playing him until he does.

Shiva
When you are in Timber, go to the Timber Maniacs building and look at the magazines on the floor.  You should receive the magazine 'Girl Next Door'.  Keep this.  When you get on to the White SeeD Ship (on Disc 3), go talk to Rinoa's friend (the one who likes naughty magazines) and tell him you have the magazine.  Give it to him for free, and he will give you the Shiva card in return.

Squall
Win from Laguna (I think)

Zell
Win this card from his mother.

Ward
Win from Dr Odine in Esthar.
More to be added.