I had played the console via store demos a few times, but the first time the PS1 really wowed me was with Final Fantasy VII. My brother's friend brought it over to our house to show it off. He was at the endgame and mostly showed us the Gold Saucer stuff. I was blown away by how many different things there were to do. Then he went outside and used Knights of the Round. Needless to say, I had never seen anything so incredible in my 8 long years on this earth.
Some time later we got our own PlayStation and obviously FF7 was one of the games I had to have for it. As my avatar can attest, it left a bit of an impression on me.
BONUS STORY: There's another incredible game that didn't win me over right off the bat. One of the demos on the disc that came with our console was a very early version of Metal Gear Solid; it didn't even have the English voice-over yet. The demo was also incredibly short. It contained only the first two screens of the game, ending when Snake enters the facility through whichever route. I thought it was kinda neat, but when presented with such a tiny slice I was given the impression that it was a serious, grounded military drama and I had less than zero interest in such a tale.
Years later (circa 2000 or so) my parents divorced, and when my mom started dating my future stepdad he gave me a bunch of his old PS1 games. One of them was that neat but boring military game with the unmistakably spartan cover art. After trying all the rest, and with no greater impetus than, "Eh, might as well," I decided I would give Metal Gear Solid a shot. I don't have to tell any of you how dramatically Kojima shattered every single expectation I had of his "boring" little game. He changed my very perception of what a video game could be and gained a devoted fan for life.