[Yora]

For the PS2, Primal is an excelent but rather obscure action adventure.

Spoiler

It has a simple third person combat mode that is used a lot and the puzzles are not that complex, but it really is a very story-heavy adventure game at the core. And I don't have a clue why not everybody knows that game! It's easily a 90+ game. It's just that damn good.
And has Andreas Katsulas as voice actor for one of the protagonists.
Gameplay Example. Not good video quality, but shows how the game plays and switches between exploration and cutscenes.

The main character is Jen, who gets attacked by a demon after leaving a night club and gets visited in the hospital by a tiny gargoyle who tells her that her body is dying and that the demon kidnapped her boyfriend, so he seperates her soul from her unconscious body and takes her with him to the Otherworld. The boss of the demon is trying to take over the whole spiritworld which threatens to destroy both worlds, so they have to journey to the four demon realms and restore the normal order. The gargoyle and Jen have different abilities and you can switch between either of them at any time, to open gates and solve other puzzles. Pretty soon Jen also gains the ability to turn into one of the four demon kinds, giving her other abilities like jumping, swiming, and super-speed. It's well written, well paced, and looks amazing on the PS2. The same graphics could be released on the PS3 and wouldn't look too bad.

Also for PS2, Shadow of the Colossus. You don't have to agree with Zero Punctuation reviews, but almost everyone should come to the same conclusion that this game is "just damn good! Damn, Damn, Good, Good, Damn, Good, Damn, Damn.
Good."
Also a kind of action adventure. It has barely any text and it's all cutscenes with subtitles. And it has no NPCs, no items, no skills, no random enemies, and it all takes place in a desert. Just 16 puzzle boss fights, your sword, and your horse. And it's incredible!

Spoiler

The plot is incredibly simple. You are a young warrior and bring a dead young woman to an ancient temple in a desert that is forbidden to enter, because there is a trapped "Being" with magical powers. The spirit might be able to restore her to life, but first the "hero" would have to destroy the guardian golems whose life force power its prision. And he might have to pay a great price to get his wish.
After the intro, there is just you and your horse. And the giant empty desert. There is no way to make the gameplay sound good, but it's an incredibly use of atmosphere that never makes it get boring as you travel to find the guardians. And then you have to kill them. But they are so huge that your tiny sword and arrows can't do anything to them, and to get to their weak points, you have to climb on their bodies. Which they are not willing to let you do.

I think as an example I take just the trailer. It shows only of the guardians very well and that one is jut the first one. Which is rather on the small side.

And just as important, get yourself a PS1 memory card and then you can play Metal Gear Solid on your PS2. Or if you have a PS3, I think the game exists for download as well.
One thing that many people dislike about the series is the massive amount of cutscenes, but if you like JRPGs that should not be a problem. The characters are mostely Americans, but it really is a throughly Japanese game in any other aspect. Then there is also Metal Gear Solid 2 and Metal Gear Solid 3. And Metal Gear Solid 4 for the PS3. I think not long ago 2 and 3 were released as a PS3 version bundled with a PS3 port of the PSP-Game Peace Walker, which is also very good from what I've heard.
All in all, the four main games alone will easy get 100 hours of what is pretty much the most pacifistic action hero there has ever been. It's at times both silly and deep. And somethimes both simultaneously. Damn, this series is incredible, it would be a crime tonot give at least the first one a try. It's just that damn good, if one is okay with the linearity and amount of cutscenes typical of JRPGs.

Spoiler

It's set on Earth in the 1990s, but there is no way it could be called the Real World. That world is a very strange one. MGS may look like a cold war spy game at first, but it is anything but that. It centers on Solid Snake, a genetically engineered one-man-army/super-spy who is reluctantly fighting an incredibly complex cospiracy that ultimately is a lot about mind controlling the masses and how ****ed up the world of super-spies actually is. And most of the villains have some kinds of magical abilities in the style of X-Men. Also, it's terribly pacifistic, focusing on avoiding enemies by using stealth and spends a lot of time with the tragic villains. Almost all of them have death scenes that are 10 minutes and longer, during which they are still talking to Snake after he fatally shot them.

Since it's so cutscene heavy, I take the intro from the third game as an example for the series incredible style.

For PS3, as a JRPG-Fan, one should get Valkyria Chronicles. It's a turn based team strategy game, but it really plays a lot like a much more complex Final Fantasy. You just have four to ten characters instead of three and you move around the map, but it's a lot of fun.
It's also a game that should be famous for being the one that does not have a single decent video on youtube.

I also dare mention Mass Effect for the Xbox 360. Technically it is a western RPG but completely different from Elder Scrolls or Fallout 3. It is rather linear and very story and dialog heavy, with huge amounts of attention paid to the characters. I also can't stand those games that have you running around in a huge world without real purpose and only banal conversations with random people you meet only once, but Mass Effect easily has as much character interaction as any Final Fantasy game. Or actually, it has a lot more! If there are games with engaging stories, Mass Effect clearly is one of the best.
The first game still plays similar to Knights of the Old Republic or Dragon Age 2 to some degree, which is one of it's weaknesses. But the second game makes it a pure third person action RPG. You do control it like a third person shoter, but compared to all the story you get, the combat parts are not actually that many. If you play it, play the female character. The male voice actor does a good enough job, but FemShep has Jennifer Hale. I loved her in Metal Gear Solid and I love here even more here.

I think this is pretty good and spoiler-free example from the second game of how most of the series plays.