You and the other player each pick three cards and then take turns placing them. Each card, when placed, "attacks" in each direction represented by an arrow on the card. If the attack is successful, any card adjacent in that direction is captured and becomes one of your cards. Whoever controls the most cards at the end wins the round, whoever wins 3 rounds first wins the game and gets to take a card from the losing player's set of three.

Attack power is represented by the letter, with E being the worst and B being the best, while defensive power is represented by the number, with 1 being the worst and 4 being the best. I'm not a big fan of Triple Triad-style minigames, but I found it was best to take one decent defensive card for the times where I had to play first, and otherwise focus on having good offensive cards - capturing other cards is how you win, after all, but the first card played in a round can't possibly try to capture anything, so its only job is to not get captured.

Attack vs. defense probabilities:
(Pretty much shamelessly copied from the official guide, which is freely available.)

As you can see, defensive power 2 isn't a big upgrade, but 3 and 4 are pretty big, as are all attack power upgrades. So, if given the chance, take D or better attackers over defensive power 2, but my advice of taking a single strong defensive card for when you have to play first still stands.

As with items from chests, Trioarch games are pretty much random in terms of what cards the other player has, but are divided into different grades.

Grade 1 players will have the following possible cards:
Dunewyrm, Ikewaise, Phyxie
Mafaleen, Ikewaise, Phyxie
Mafaleen, Frofaleen, Phyxie
Mafaleen, Dunewyrm, Phyxie
Mafaleen, Frofaleen, Ikewaise
The first one of these you can find is Katina, in the City of Strife.

Grade 2 players will have the following possible cards:
Purple Widower, Frofaleen, Mechaguard
Musch, Frofaleen, Mechaguard
Musch, Aquacyte, Mechaguard
Musch, Mafaleen, Purple Widower
Aquacyte, Mafaleen, Mechaguard
The first one of these you can find is João Gonçalves, in the City of Strife.

Grade 3/4 players will have the following possible cards:
Gaiacyte, Fallan, Knight
Rook, Fallan, Knight
Rook, Papierma, Knight
Purple Widower, Gaiacyte, Knight
Purple Widower, Gaiacyte, Papierma
The first one of these you can find is Talaan (or EpicManeuver), inside the Communal Hall.

Josh Turney in HC-2 and {Beta} Marcus are the only level 5 players, so make sure you leave at least one of the two "alive" until you've gotten the cards you want from them. They will have the following possible cards:
Rook, Papierma, Crystalline
Rook, Papierma, Safeguard
Nityt, Papierma, Safeguard
Nityt, Papierma, Rook
Safeguard, Papierma, Rook

Duncan (at the top of Tantaluz Tower) will always have the following cards:
Purple Widower, Duncan Dellamorte, Nityt

Dunewyrm - Dropped by Dunewyrm (1/50 chance) in Level 1: Desert or Arena Level 1 or carried by Grade 1 Trioarch players. Three of these are given to you outside the inn in the City of Strife, and they are sold by the merchant in both Hax Compleck and HC-2.




Ikewaise - Dropped by Ikewaise (1/50 chance) in Level 2: Forest (but there's only 3 and they don't respawn) or Arena Level 1 or carried by Grade 1 Trioarch players. One of these is given to you outside the inn in the City of Strife, and they are sold by the merchant in HC-2.




Phyxie - Dropped by Phyxie (1/50 chance) in Level 5: Magmacave or Arena Level 3 or carried by Grade 1 Trioarch players. One of these is given to you outside the inn in the City of Strife, and they are sold by the merchant in HC-2.





Mafaleen - Dropped by Mafaleen (1/50 chance) in Level 6: Ice Fortress or Arena Level 3 or carried by Grade 1 or 2 Trioarch players.






Frofaleen - Dropped by Frofaleen (1/50 chance) in Level 6: Ice Fortress or Arena Level 3 or carried by Grade 1 or 2 Trioarch players.






Mechaguard - Dropped by Mechaguard (1/50 chance) in Level 9: Warehouse or carried by Grade 2 Trioarch players. One is found in a chest outside the Communal Hall.





Musch - Dropped by Musch (1/50 chance) in ZoKi or carried by Grade 2 Trioarch players. One is found in a chest in the City of Strife.






Aquacyte - Dropped by Aquacyte (1/50 chance) in ZoMa or Arena Level 4 or carried by Grade 2 Trioarch players.






Hollow - Dropped by Hollow (1/50 chance) in KiGe.







Gaiacyte - Dropped by Gaiacyte (1/50 chance) in KiGe or carried by Grade 3/4 Trioarch players. One is found in a chest in Fort Leonhart.






Fallan - Dropped by Fallan (1/50 chance) in AlGe or Arena Level 5 or carried by Grade 3/4 Trioarch players.






Knight - Dropped by Knight (1/50 chance) in MaGe (but there's only 3 and they don't respawn) or Arena Level 5 or carried by Grade 3/4 Trioarch players. One is found in a chest in Fort Leonhart.





Rook - Dropped by Rook (1/50 chance) in MaGe (but there's only 2 and they don't respawn) or carried by Grade 3/4 or 5 Trioarch players.






Papierma - Dropped by Papierma (1/50 chance) in ZoTz or Arena Level 5 or carried by Grade 3/4 or 5 Trioarch players.






Crystalline - Dropped by Crystalline (1/50 chance) in ZoTz or carried by Grade 5 Trioarch players.






Purple Widower - Dropped by Purple Widower (1/50 chance) in StromIV or carried by Grade 2 or 3/4 Trioarch players or Duncan Dellamorte.






Safeguard - Dropped by Safeguard (1/50 chance) in City of Strife or Fort Leonhart when you hack most guards or certain players (10 in City of Strife, 9 in Fort Leonhart though one is only accessible on a NG+) or Arena Level 6 or carried by Grade 5 Trioarch players.




Duncan Dellamorte - Carried by Duncan Dellamorte.







Nityt - Dropped by Nityt (1/50 chance) in KiTz or Bug Dungeon or carried by Grade 5 Trioarch players.






Guardian - Received for participating in 100 Trioarch matches, win or lose.