In the PC game, players quest for the Longshanks Hammer, which will give them control over Dragonkind and try to claim it before evil sideshow owner Silus Copperbone, with stars being awarded for the Five F's (Fieldwork, Foresight, Forwardness, Frankness and Fatality avoidance) and dragon designing features available.
In the special NDS edition of the Dragonology game the DS becomes an interactive book, allowing the player to care for a Dragon, explore the world, discover dragons and duel while exploring Dragonology. The DS edition appears to be more popular and second hand editions are still available.
However, as few people have DSs, I would recommend downloading an emulator such as the excellent DeSmuME and a Dragonology NDS ROM - the one at http://romhustler.net/rom/nds/dragonology-eu-m6 appears to work and be virus free but beware of malvertising on the site.