Some of my favorites and why:
Final Fantasy IV/II(us):
It's mostly a nostalgia factor as it's the first JRPG I *really* got into. I'd tried Dragon Warrior and the original Final Fantasy before but had never gotten far before I got bored. However Final Fantasy IV (or II as it was known to me at the time) had a story and presentation that at the time was very cinematic and just captivated you and sucked you in. I was 12 when I first played it, and yes other JRPG's have come out with better characters, better combat systems, better stories.. but for it's time.. that was an amazing RPG. I still get a bit choked up in a few key moments during the game's plot.. the opening, when you see the Red Wings raid Mysidia, that epic 16 bit score, you see a dark hero, questioning his actions and loyalties.. then.. when that dark hero becomes a Paladin.. with that fanfare.. I get goosebumps thinking about the first time I heard the fanfare and saw "Cecil became a Paladin!".. and finally, at the end, when your friends all send you their prayers to give you the strength to take down Zeromus.
Final Fantasy VI/III(us):
Yet another amazing story and presentation, this one stands out as being one of the best aging games I've ever had the pleasure of playing. The graphics can still be appreciated, the score is beautiful, the story amazing (it's a story in which you FAIL to save the world, the basis of pretty much every other game in the genre, you FAIL, and watch the world get ruined, and then go back to get Kefka to save what's left), the cast has very distinct characters that are likeable, and probably the most evil villain I've ever squared off against, Kefka.
Secret of Mana:
This game.. was awesome.. for it's time it was one of the first RPG's that had multi-player capability, you could go hack and slash with 2 of your best buds and it was a great time. This game introduced my friends to the RPG genre, which prior to it, they thought all RPG's were boring menu driven games.. not this one though, it was all action. This game also turned me on to the action RPG subgenre, which I prefer over menu based games these days.
Tales of Phantasia:
The first entry in my favorite jRPG series, this game was a technological feat for the SNES, it's a real shame it didn't come over to the US on the SNES and only the watered down GBA port made it overseas. Voice acting (on a SNES Cartridge!), a big world and huge quest with lots of side quests to go on, the synthesis system (well, using rune bottles on items to make new items), and especially the COMBAT SYSTEM just blew everything away. This game and all of the Tales of.. games utilize a combat system that feels kinda like an arcade fighting game. Hacking and slashing, using special moves by inputting button combinations, combos, blocking/defending, it was great...
Chrono Trigger:
Hey, what do you know, some menu driven jRPG's can still be good, this is a prime example, the double/triple tech special system was awesome, time travel is awesome, multiple endings are awesome, New Game+ is awesome, this game pioneered the multiple endings and New Game+ that are now staples of the genre.
Okay and to throw in my current favorite jRPG and have something other than "the classics":
Tales of Vesperia:
This is the Tales of formula... perfected as of today's standards.. beautiful anime style visuals, a freely moving 3d field combat system with the additions of burst artes and fatal strikes, encounter links.. the combat system has never been better. I haven't finished the game yet (just finished Zaude) so I can't comment on the entire story so far, but I'm really enjoying it.