Pokemon Red & Blue (Game Boy, 1998)

Yes, Pokemon.  Before the trading cards, the merchandise, the television show, and the inevitable backlash, there was a revolutionary pair of RPGs for the handheld Game Boy, which were designed and programmed by only four people.  Game Freak was a tiny Japanese game development company looking to make a smash hit.  Final Fantasy VII had popularized the RPG for the older crowd of gamers, so they sought to create something that younger would be able to understand and enjoy.
Pokemon Red and Blue (1998) take full advantage of its Game Boy hardware.  By this time (even in Japan where the games were released three years earlier) the Nintendo 64 was out, and Game Freak might have only had the resources to develop a game for the 8-bit handheld.  Therefore, the game has a classic, but still distinct, art style, and classic RPG design.  Where Pokemon differentiates itself from the others is its deep party customization and story.
The story of Pokemon is great because it doesn't come with any pretenses and doesn't take itself seriously.  Where most RPGs were stories of legendary heroes having to save the world from some kind of apocalypse, Pokemon is about a young boy who simply wants to become the master of, well, Pokemon.  What is a Pokemon?  Why, they are the creatures that inhabit the world of this game, which a Pokemon trainer (namely, you) can catch and train to battle for you.  There are 151 of these creatures to capture, giving the game virtually limitless possibilities for customization.  You start by receiving your choice of one of three "starters" from Professor Oak, whose grandson is your rival.  With your first Pokemon, you set off to capture more, train them, beat the eight gym leaders, and then tackle the Pokemon League, four masters of Pokemon that you must face in succession.
Within its own universe, this all makes logical sense.  You catch Pokemon by using Pokeballs, which convert the living creatures into computer data which can be transported anywhere.  Pokemon battles take place between trainers, where you engage in one-on-one battles.  Each Pokemon can learn four moves, which are learned by leveling up or using TMs (technical machines).  You can carry only six Pokemon with you, and all subsequent ones you catch are sent to your PC.  The reason that there are two games are that in each game, there are a different set of Pokemon you may capture, and the only way to get some of them was to trade via link cable with a friend who had the other version of the game.  This system has persisted to this day.
One thing I noticed upon my recent re-visiting of Pokemon was how dark it can get in certain parts.  There is a gang called Team Rocket who appear from time to time (this was before the characters Jessie and James made them silly) who do some cruel things with Pokemon.  They illegally capture, trade, and even kill them.  The spirits of dead Pokemon go to the Pokemon Tower in Lavender City (whose creepy music was rumored to have caused child suicides in Japan).  On Cinnabar Island, there is a burnt laboratory where the Pokemon clone, Mewtwo, had gotten out of control and destroyed it, killing the scientists within.  I said this was a game for children, right?
The Pokemon franchise would explode in the States over the next few years, but its popularity was short lived.  By the time the third generation of games came along, fans had grown out of it.  I, myself, was a huge fan of Pokemon when I was a kid.  I watched the TV show, played the games, and collected the cards (I never actually played the card game).  The last entry I owned was Ruby.  The series didn't die though; it is still going strong to this day and continues to set sales records.  I was worried that the original games would feel dated, but like so many of the other favorites from my childhood, Pokemon has aged like fine wine.