The thing that draws me to Pokemon is the depth of the game - Pokemon Pearl has so far lasted me almost 900 hours over the span of almost 2 years now, and still going just as strong as it was from the beginning. The only game I've earned a longer play time for is Super Smash Bros. Melee at 2,500 hours (and Melee is still going strong too now that I've quit Brawl,) but I digress.
As for what is actually in the games that makes them so fun...well, everyone who knows me here knows I specialize in competitive battling, so it won't come as any surprise to anyone that the battling is what I find the most fun. The thing that makes the battles so fun is their depth - again with the depth, but it is true. You have EVs/IVs/Natures that add incredible strategic depth to training and breeding Pokemon, and then the battles themselves end up being like chess games, where you have to plan your moves and predict what your opponent's moves will be in order to win. No other game provides such a deep and unique multiplayer experience as Pokemon does, and that is what makes it special.
Also, the sheer number of different things you can do within the game makes it special. You have contests, mini-games, you can focus on collecting Pokemon for your 'Dex, focus on collecting shiny Pokemon, could go on and on here, but I won't. There's a niche for pretty much everyone within the game.
Finally, going back to what Silver Magpie said about the forum making the game fun, I agree 100% with that. Pokemon is great at bringing people together; I first met my best friend through Pokemon, and I've made a lot more friends through the game both in real life and here in 'Charms. Also, the numerous Pokemon events that are held are a fantastic way for fans to meet up, trade, battle etc. My first event was the Hanley Darkrai event which was held last August if I remember correctly, and it was an absolutely amazing day out for me.
The game almost blurs the line between its world and the real world in that respect, something that very few, if any, other games can really claim to be capable of.
So...there you have it.