As someone who has played all but one generation of Pokémon, I can definitely say that this generation is my favorite of all of them in many ways. From the get-go, two things stand out. The first is that, obviously, this is the first main series game to be fully in 3D, and by that I don't even necessarily mean the type of 3D that the 3DS deals in. I mean that this is the first time that the full game world, all the characters, and even the Pokémon themselves during battle are three-dimensional rather than just flat, 2D sprites. So, even if you're the type of person who keeps that 3D slider off at all times, this generation visually far exceeds it predecessors.
The second is that you have more freedom with your character design this time around. In addition to being able to play as a boy or a girl, you have three skin tone and hair color combinations to choose from. However, that's just at the start of your game. As you progress, you will come across a number of boutiques selling clothes with which you can customize your character's look, and one city offers a salon for changing hair color and style. A particular NPC (non-player character) even offers you a set of colored contact lenses at one point, so you really do get a lot of options.
Then, there's the gameplay, which has added numerous mechanics. Some are more subtle, such as the facts that your Pokémon now get experience even when you capture a Pokémon and that each Pokémon that participates in a battle receives the full amount of experience (rather than that amount divided by the number of your Pokémon who participated). Others are more obvious and make raising a good team more accessible. Super Training allows you to see and to train up specific base stats for each Pokémon, such as Speed, HP (Hit Points), and Attack. This is done via minigames and can be done to whatever extent you choose. Otherwise, those stats can be raised through battle as with previous generations (though this is a much more precise and hidden process that requires a guide if you're seeking to train specific traits). There's also Pokémon-Amie, which allows you to interact directly with your Pokémon, petting, feeding, and playing with it. This raises its affection, which, when sufficiently high, allows it to survive battle-ending hits or to ignore status effects out of sheer love for its trainer.
Of course, it would be remiss not to mention important changes to battling. For the first time since the second generation (think Gold, Silver, and Crystal), there's a new type. This type is Fairy, and, as with the last introduction of new types, it changes a number of matchups. There are a number of places where you can find the updated type-matchup charts online, but there are other changes as well. Steel is no longer resistant to Dark or Ghost; Electric-types cannot get the Paralyzed status; Grass-types are not affected by powder moves; and Ghost-types are not affected by trapping moves or abilities. These all add extra nuances to the already strategy-heavy battle scene.
Trading and online capabilities got a boost as well. As with the last generation, there are powers (called O-Powers rather than Pass Powers this time around) that can be utilized on yourself or on others to benefit the user in some way, and these can be leveled up. Battles and trading can be done online. Trading in particular has a couple of new options, Global Trade Station and Wonder Trade. Global Trade Station allows you to put a Pokémon up for trade under certain conditions. Uploaded Pokémon can be searched by anyone, and if they are willing to meet the conditions of the uploader, they can orchestrate the trade. This doesn't require you to stay connected to the Station, so you can upload a Pokémon and then come back later to see if you've gotten your requested trade. Wonder Trade, on the other hand, is simply random trading over the Internet. Trade wisely, but the idea is that you offer a Pokémon for trade and receive a Pokémon from some random other person using Wonder Trade. It can be a decent way to acquire Pokémon you don't have without needing to have people you know who play the game or needing to deal with the particulars of the Global Trade Station. Just remember that there will be both people trading great Pokémon and common, easily-acquired ones.
There's also the concept of Mega Evolutions, which were part of the ad campaign for these games. I won't go into this, since it forms part of the plot, but it's a very effective and interesting mechanic, especially if such an evolution exists for any of your favorites. For fans of starter Pokémon, you get one of the starters from the first generation (Squirtle, Charmander, or Bulbasaur) early in the game.
No matter how you look at it, this game brings a lot of new stuff to the table. My one gripe is that, compared to the last generation, where your rivals felt more developed and various characters (particularly gym leaders) were more involved with the plot, the story in this one feels less developed. However, the shorter post-game story is engaging, and the high stakes and occasionally somewhat-dark subject matter that are part of the plot with the antagonistic team help to alleviate this issue. Just be sure to talk to everyone and to experience all of what each city and building offers to get the most out of it.