With games like Baldur's Gate, Neverwinter Nights, Kingdom Hearts, Final Fantasy, Paper Mario, Oblivion, Dragon Age, Jade Empire, and more (though some of these could probably be disputed), my favourite genre has to be the RPG, or Role Playing Game, for the unenlightened. This was probably obvious, especially considering the amount of Dungeons and Dragons I like to play. There are two main reasons for why RPGs are awesome:
The RPG is the realm of video games that lets you, well, role play. Instead of in many other games where you control a protagonist who you have no connection to all and has no personality, an RPG lets you care about the character you're playing. The best RPGs let you control the fate of the character, taking them down a number of different moral paths. This adds a layer of ethical quandary and intelligent decision-making to the game, and lets you forge the character into a number of different options.
Neverwinter Nights and Oblivion are probably two of the best examples here, where the gameworld allows you to be whatever type of person you want to be, and Oblivion takes this the farthest.
Some are more flexible than others, Jade Empire being the least flexible on this list perhaps, or Kingdom Hearts, where the character is taken along a straight story path. Deviation and exploration is allowed, but the character will not change much due to your choices. JE at least has a scale of good/evil which affects some abilities.
I really like this aspect of RPGs. For a start, its liberating to have so much freedom in a game, and to be able to make choices. I like free will... On top of that, I like to see the different parts of the game which you can only see if you are only evil or only good, like Thieves Guilds and so forth. Finally, its a great way to get to know yourself. I find over and over again that I simply CANNOT play an evil character. I manage it once in a while, but generally I really do not like making those nasty choices. Turning down an orphan's plea for food or killing the messenger who brings you bad news is difficult, and that's somewhat of a rewarding experience to discover your own morality, even in pixelated form.
The second part of RPGs that I like is of course the character personalisation. Not just forming their moral and general life choices, but their physical form and abilities as well. Rolling stats, choosing abilities, determining exactly what kind of person you will be playing as for the rest of the game is great fun, and there's nothing better than Leveling Up. Seeing your character get stronger, and tweaking them to be excellent at what they do, be it diplomacy, fighting, theivery, magic, or a mixture, is a blast.
So generally, RPGs are fantastic because they allow for indepth and continuous character personalisation and development. Certainly all RPGs allow for the physical development side, if not the moral side. I enjoy the story driven action of the games, the difficult choices it throws at you (Dragon Age was particularly good here), and the challenge it puts to you at all times: Who will you be?
As a side note, its worth pointing out that Chrono Trigger is somewhat unique in that you can TRY and be a mean person, but they shut you down for doing so fairly quickly. You're put on trial early on in the game, and if you have been a mean person so far, you will be executed. Game Over. Harsh, but perhaps somewhat more realistic in some ways than, oh-I'll-just-use-this-glit