N64/PSX, and it's not even close.
As a kid, I liked playing videogames as much as the next kid. I enjoyed playing Super Contra on the NES, Super Mario Allstars on SNES and Sonic on Megadrive. But it was something we did besides playing Lego or playing in the woods. Not a serious hobby.
One day however, our local toystore had an N64 demo setup, running video's of Super Mario 64 and Wave Race 64 and I remember being awed by those visuals and thinking by myself 'videogames can be like this?'. Similarly, I remember seeing the TV-ads for things like Final Fantasy 7 and Ocarina of Time and being absolutely amazed by it. Not long after that, my brother got a PSX and I got an N64.
To me, it was the only time a new gen actually felt like a revolution. Obviously, it's also the gen that has aged the worst and I consider most of these games unplayable nowadays, especially since we're in PAL-land and most games run at like 17fps, but whatever, at the time this stuff was truly nextgen. No other transition ever gave me that feel again.