Hard to pick one, so I'll just talk about my favourite series and my favourite games within them.
Lunar: The original Lunar games (Silver Star Story and Eternal Blue) were released on the Sega CD, but I played them for the first time when they were remade for PS1. To this day, Lunar sets the standard for what I like in an RPG, both in terms of characters, plot, and gameplay, as I think Lunar does great in all three, specifically Eternal Blue. Although the series kind of fizzled and died some time ago (the GBA remake of SSS was rather lackluster, unfortunately, and don't even bother with whatever is up with the weird DS game that doesn't even deserve to carry the Lunar name), the PS1 and Sega CD games are amazing and highly recommended.
Chrono Trigger: Another amazing RPG, Chrono Trigger manages to withstand the test of time and still be one of the greatest RPGs ever made. It makes good use of time travel and the gameplay is a lot of fun. It also manages to set a very creepy atmosphere and I love its interpretation of the various time periods and settings; the game really makes you feel like you're a part of the world, and it doesn't need crazy graphics to do so, just wonderful descriptions and artwork and music. I replay this game every couple years just so I can remember how awesome it is. This is one of the few games that I think every gamer should play at least once, and especially game designers. In short, <3
Golden Sun: Finalizing out my trio of "awesome RPG series" is Golden Sun, a two-part series on the GBA with a DS sequel as well. Golden Sun manages to be rather different from both of the other two series I love mentioned above, but is still an amazing game with an enticing story and excellent gameplay. One of my favourite things about the game is the Djinn system, which changes your stats, abilities, and even class based on how you use them, giving birth to a wide variance of gameplay. The game itself also has a much different difficulty than typical RPGs, and changing strategy and tactics is much more efficient at playing the game well and progressing than boring level grinding. The DS game isn't as good as the 2 GBA games, but they're all fun.
Fatal Fame: Fatal Frame is the best horror video game series I've ever played, seen, or heard of. The gameplay mechanics (using a special camera to take pictures of ghosts) bring you right up close to the action, which makes things much more intense, and the atmosphere is made up wonderfully: use of colour, camera angles, music (or lack thereof), sound effects, controller vibration all combine to make a chilling experience you won't soon forget. I'm kind of a horror junky, and this is the only medium (game, movies, books, whatever) that has ever actually, honestly, scared me. Most people say the second game is the best of the series, and while I do think it's good, I think personally prefer the first.
Ace Attorney: The AA series is amazingly fun and witty, something you might not expect from a game that stars a lawyer trying to defend their clients from murder charges. The games basically play like a more-detailed visual novel, as you investigate crime scenes, interview witnesses, and then play in court to try and find your client innocent and find the true killer. The characters are wonderful, the stories interesting, and there's an overall plot between the games, too that ties everything together. Also worth mentioning are the spin-off games featuring the main prosecutor from the first games; they feature characters from the main series AA games but different (though similar) gameplay and their own storyline. The first 3 games are soon being released as a combo title for the 3DS, so soon no one will have an excuse to have not played them! My favourite game in the series is the 3rd one, Trials and Tribulations, though the recent 3DS release, Dual Destinies, takes a close second.
Fire Emblem: FE is a turn-based strategy RPG series developed by Nintendo. Even people who haven't played the games are often familiar with the characters from SSB: Marth, Roy, and Ike. FE started out on the NES and has since been released on many Nintendo consoles, both regular and portable. The series made its debut outside of Japan with the 7th game in the series which was called simply "Fire Emblem" in those countries. Gameplay changes and tweaks happen in each new game, but the core gameplay remains the same as you build an army of soldiers and mages and use tactics to win battles and progress. My favourite game in the series is actually the newest, FE: Awakening, which was released on the 3DS last year, though otehr top contenders are Seisen no Keifu (4th game, for the SNES, available as a ROM with translation patch for those who don't know Japanese) and the first one that was translated to English as simply Fire Emblem on the GBA.
Legend of Zelda: LoZ is kind of an interesting series because it's so long and it's changed so much from its origins, and yet so many core aspects of the games have stayed the same. It's probably the most popular game franchise that I like (either this or Pokémon
) but I can understand its popularity because it's legitimately fun and has good gameplay. I can't help but miss the old 2D Zelda games, but the newer 3D ones are a lot of fun as well. Majora's Mask is probably my favourite, and I'm crossing every one of my fingers constantly for a remake or sequel on the Wii U, like has been hinted it. That said, the Oracle games are also a very close second; the dual storyline is cool, and I rather like the 2D aspect of earlier games, since it simplifies some things.
Pokémon: Don't need to say much about these games since everyone here knows them.
It's hard to pick a favourite Pokémon game (or even generation) but either 3rd or 6th gen has to be my favourite; 3rd gen because of contests and the hilarious plot with Teams Magma & Aqua; 6th gen because of Wonder Trading and all the awesome new Pokémon and the upgrades to the world and such.