The pokemon game system/paradigm is highly unique among the world of games. The main character (you), lives in a world where humans called trainers collect animal-like creatures, called pokemon, and use them to help in their travels and exploration. This primarily involves using pokemon as combatants and defenders, against other wild pokemon, or in organized battles with other trainers.
Care/Harm: As can be expected, care/harm is a controversial issue for the pokemon franchise, which has, in fact, recently been targeted by PETA for espousing animal cruelty and such. Though it is oft repeated in game that humans and pokemon live together in peace, it is at times difficult to imagine pokemon voluntarily submitting to a life of servitude and forced combat.
Fairness/Reciprocity: The game does not in and of itself implement fairness enforcing systems, explicitly, though it can be said that work and effort are rewarded by improved performance, and that with the capacity for trading implemented in later game editions (which must be approved by all sides), there are implicit fairness rules built into the interactions.
Loyalty/Ingroup: The player does not have the opportunity to join groups, and is in fact often countered by antagonists who are members of specific recurring gangs or groups. This in a way showcases the negative aspects of the group or mob dynamic, focusing instead on the power of the individual player as the moral agent.
Authority/Respect: The game heavily incorporates ideas of authority and respect, a common theme in level based games. Nearly everything revolves around power hierarchies. The player leads his party of pokemon. Various trainers work at gyms and are subservient to the gym leaders, who are themselves under the Elite 4, who fall below the regional champion. As the player increases in strength, they can challenge these different levels of authority/skill, and rise in the ranks
Sanctity/Purity: Ideas of sanctity/purity are, unlike in many games, fairly important. As a pokemon trainer, taking care of and training your pokemon well involves carefully monitoring their actions, their foods, what kind of creatures they fight against, etc. all of these things can lead to drastic changes in the pokemon’s development.
The game has aspects which can easily appeal to any kind of player. For socializers, there are the options for communications, trading, and battling over wifi. For killers, there is the entire battle system itself. For achievers, there is an entire hierarchy to climb, and hundreds of different pokemon to discover and capture, and for Explorers, a whole world to discover, which continually expands with the addition of new games.
As far as the games other people have discussed, I would probably enjoy just about all of them, except perhaps Angry Birds, League of Legends, or Kid Down the Stairs. I tend to prefer either more complicated or role-playing games which allow for character growth, development and exploration, or short games which can be played directly with friends, preferably in person (head to head). I am not a fan of single player killer focused games like first person shooters, but I do enjoy multiplayer fps’ for the socialization/game aspects.