I loved how they shifted the focus off of what happened around the characters and focused on what happened between them. It made me feel like I got to know them as people. The last game that did that before this was FFVII.
For ps2 it would have to be the entire Xenosaga trilogy. Seeing the development of Kos-mos and Shion's relationship was heart warming at times, and had me at the edge of my gaming chair at others.
For ps1 I actually have two. The first is pretty much everything about FF7, the second is the amazingly well done story of FF8. I actually liked FF8's story better than FF7, but the junction system was a little to complicated for me, and I didn't like having my spell options limited by the effects they would have on my character's stats.
For PS2 - MGS 3 - The sniper fight with The End, loved it! Actually all the boss fights were pretty cool. LOVE THAT GAME!
PS3 - so many to choose, but I will go with inFAMOUS 1. I didn't really know what the game was about, but being able to jump off tall buildings and land on the ground with no harm (no water), fire electric sticky grenades (nice) and climbing that tall junk structure (seemed to take forever). Sidekick Zeke was also funny.
I'm not too sure about PSP. Just generally, it would probably be trading in my DS for it and starting Daxter for the first time, as well as the various deleting and copying and whatnot I had to do to play demos on my meager 32MB memory stick when I first got it.
For PS3, there's a couple. First, (inFamous spoiler)
was when your girlfriend dies when you're good Cole. Second would be the good ending of Bioshock. Third would be playing the first level of Resistance 2 at a demo kiosk. I've yet to play anything that struck me as being that epic.
God of War II (PS2) - 'The Crossing of the Lowlands' is still one of my favorite moments in ALL of gaming. That section of GoWII where you have only grapple points to advance while the columns are crumbling around you... yeah, THAT section.
Can't really think of any other particular moments as memorable as that one.