A myriad of answers before me. Let me add to the mix.
Pokémon: RBGY, without a doubt. I'm not a Genwunner, but I grew up with these Pokémon from a very young age, and they've had a profound effect on me and the way I deal with the franchise as a whole these days.
Starters: Again, probably RBGY. Bulbasaur was my first ever starter Pokémon, and just seeing it's art brings back fond memories, and all three have great, simple designs that will always be loved by the fans, both new and old. Coming in at a close second would have to be the RSE starters, Treecko, Mudkip and Torchic. I originally started Ruby with a Mudkip, but having become frustrated by not being able to get past a certain point in the game and thinking that Pokérus was a bad thing, as well as thinking I could try a new save file and not save over my current one (which worked, but I saved over it myself anyhow), resulted in me playing through with a Treecko. I got to the E4, and with Treecko now a Sceptile at around Lv80, I had to grind both there and at the Trick House. Upon reaching Lv93 and learning the TM Dragon Claw, we demolished the League, just me and my Sceptile.
Legendaries: This one's very close, but I'd have to give DPPt the slight edge over BWBW2 here. The most legendary Pokémon ever introduced in a region, and whilst some were kind of strange, I loved the focus on Dragon types, and was proud to have stumbled upon Turnback Cave by myself, getting through and getting a Giratina as the first between me and all of my friends. Arceus and Darkrai have great designs too.
Region: Kalos. Say what you like about the story and the characters, but this was very easily the most beautiful region we've visited in the series thus far.
Protagonists: I'm going to go with GSC, Gold and Kris, specifically Gold. I watched the Pokémon Chronicles special "The Legend of Thunder" and loved the design of both Jimmy and Typhlosion, to the point I went and named my character in Soul Silver after him and started with a Cyndaquil. You rock, you rule!
Rival: GSC, Silver. Not that there's much to argue about this - only Silver and Blue were the rivals we actually hated, and Barry the only one I enjoyed battling other than these two. A dark, mysterious past and constantly brooding, it was great to see this character quite literally evolve through the game's course.
Professor: Oak. Gave me my first Pokémon, and well...Oak.
Gym Leaders: DP. Great variety, I can't remember seeing any of these as a walkover.
Elite Four: Probably RSE, simply for the memory of grinding with my Sceptile. Drake was no slouch, either.
Champion: DP, Cynthia. Easily the most difficult Champion battle, and it felt like we were truly facing a friend/mentor figure.
Evil Team/s: BW/BW2, Team Plasma. The greatest evil team in the Pokémon franchise, these guys never slacked and actually accomplished their goals. You know, before Ghetsis revealed to N that he was just being used and we showed up. N and Ghetsis are easily the most memorable evil team members, and it was so refreshing to have an evil team who's goals were in kind of a grey area.
Plot: BW/BW2. Gen V mastered their characters, the plot and the evil team.
Music: Either RBGY or RSE. I don't think those require explanation.
New/Improved Mechanic: Now this is a question I've had to think really hard about. Hmm...probably the introduction of breeding in GSC. We seem to quickly forget that this has a massive effect on the games in a variety of ways.