What's the name of the card game in Final Fantasy 8?

Triple Triad was designed by Hiroyuki Ito, the battle designer for FFVIII, and wasn't based on a specific card game as far as I know.
It did however spawn a physical version, described here
Also check out Triple Triad Online

It is called Triple Triad.

The Triple Triad (トリプルトライアード, Toripuru Toraiādo?) is a popular card game in Final Fantasy VIII. According to the Final Fantasy VIII Ultimania the card game was created by a psychic named Orlan who modified fortune-telling cards for use in a game, coining the official name "Triple Triad."[1] Triple Triad was initially played among soldiers, but spread to the common people and by the time of Final Fantasy VIII's events, the game is extremely popular among all age groups.

Triple Triad is played on a three-by-three (3x3) square grid of blank spaces, where cards will be placed as the game progresses. The cards depict various characters, monsters, and bosses from the game. Each card has four numbers (known as Ranks) placed in top left corner; each number corresponds to one of the four sides of the card. These numbers range from one to nine, the letter A representing ten. On the top right of the card there is sometimes an elemental symbol representing the card's element. The elements are Earth, Fire, Water, Poison, Holy, Lightning, Wind, and Ice.

It is not based on any real game. Bandai-Namco did make a physical version at one point, but it was a limited, Japan-only run.

For those finding this question in 2017+, Triple Triad is now available as part of the official Final Fantasy Portal App from Square Enix, found here. It has cards referencing characters from all mainline FF games and lets players compete online.