These are official trading cards, published by Enix and a company called Tenky in 2000 & 2001.
The cards ranged from the first DQ game, up through DQ VII, including DQ Monsters and DQ Monsters 2. The final set was released right around the time DQ IV was released for the PSX.
The card game follows typical CCG style. There was a base set, and 5 expansions, with the occasional fixed preconstructed deck release, and a fair amount of specials and promos. In all the sets, each pack had 1 rare card, normally a foil card.
Across all the sets, there are 817 cards. There were 5 "Secret" cards - which were found in sets 2-6 at a rate of 1 in 4 boxes of packs or so (that means about 1 in 100 packs). These were Darkdrium, Orgodemir, Ramia, Dimensaur, and Kagebou (the latter three are the secret monsters from the DQ Monsters series). These "Secret" cards were not on any official card lists, so they were truly something special if you got one and didn't know they existed. These 3 secret monsters were also re-printed with slightly different graphics and made up a set of 3 more promos from magazines. Finally, there were 24 promo cards from various sources over the years the game was in print.
Cards 001-141 made up Volume 1. These were sold in booster packs, as well as a starter deck which had enough cards to start you playing. It had a nice assortment of heroes from the series. Not every rare was a foil. Cards of note in the set were the Hero from V, VI, and VII, Bianca, and all the main characters from VII.
Cards 142-189 made up a small set of cards which were not sold in packs. Not sure what the history is here, but I normally see these called "EX". From this point on in the sets, all the rare cards were foil.
Cards 190-204 made up a special set of powerful monsters and bosses. This set included Dragonlord, Hargon, Malroth, Baramos, and Zoma.
Cards 205-304 were the second full set of cards to be sold in packs, Booster Volume 2. Cards of note included a lot of foil dragons, as well as foil versions of the hero from DQs I and VII.
Cards 305-364 were only available in two preconstructed decks. Included in these two decks was the only copy of DQ VI's boss, DeathMore.
Cards 365-465 made up Booster Volume 3. This set was a celebration of the release of DQM II in Japan. Cards of note in the set included Darck (boss from DQM II), a Grandslime, and the suped-up versions of DracoLord and Malroth that were introduced in DQM II.
Cards 466-523 made up Booster Volume 4. This was only a half-sized set, with the packs containing about half of these new cards, and half of the "EX" cards reprinted.
Cards 524-583 were only available in a second set of two preconstructed decks. These featured Mudo and Estark.
Cards 584-685 were Booster Volume 5. This set had a nice assortment of "powerful looking" monsters. Super rares included the Zenithian Castle (from DQ VI, when you fight it), a suped-up Zoma from DQM II, and an awesome card that had both the 1st and 2nd form of DracoLord printed on it - with 2 sets of stats. It also had 5 heroes - female warrior from III, female hero from IV, male merchant from III, hero from DQ I, and Barbara from VI.
Cards 686-785 were the final set released, and came out around the same time as DQ IV for the PSX. The set included Necrosaro, along with his human form.