The titles that stand out to me the most are:
3D:
Tobal 2- This was absolutely insane in 1997. The high resolution and silky-smooth 60 FPS made this game look and animated so much better than all the other 3D fighters out at the time. A lot of people are going to cite Tekken 3, but this was a way more impressive game. Shame it never came Stateside.
Slap Happy Rhythm Busters- This cel-shaded 2.5D fighting game had some awesome character design and a fun dance game-like finishing mechanic. Looked really good at the time.
Vagrant Story- I didn't play this very much -wasn't for me- but it was an absolute head-turner when it first came out
Bust-A-Move(Groove) Dance And Rhythm Action: Probably not the best of screen-shots, but this motion-captured dance battle game was amazing when it came out. I couldn't get enough of it and its sequel.
Ridge Racer Type 4: A lot of people might go for Gran Turismo over this, but the night stages in this game were really awesome.
Metal Gear Solid: Had absolutely stunning presentation at the time.
2D:
Street Fighter Alpha 3: This was the first time that Capcom ported a new fighting game to the PS1 only and not to the Saturn as well. Of course, that would change later -the game made it to the Dreamcast and then the Saturn after a period of exclusivity- but this port, while certainly not perfect, was such a step up from previous ports that I didn't really miss the near-arcade perfection on the Saturn. They also added some extra effects.
Real Bout Fatal Fury Special: Dominated Mind: This was a PS1-exclusive 'upgrade' to Real Bout Special. SNK changed the game up, ditching the line-sway system, which in turn allowed for more animation in the sprites and added animation in the backgrounds, not found in the original Neo-Geo version.
JoJo's Bizarre Adventure: This game had pretty much static backgrounds, but was still pretty impressive and was the PS1's only CPS3 port. Like SFA3, it had extra effects not found in the arcade version, IIRC.
Castlevania Symphony Of The Night: This game re-uses assets from the 16-bit Rondo Of Blood, but the huge stage areas and amounts of enemies in them were pretty impressive at the time.