Probably GBA or PS2. I was a teenager and started working part time jobs, so I had money to buy the games I wanted. I was also getting a bit older, so my parents were relaxing my restrictions a bit. I couldn't play RPGs as a kid since apparently anything having to do with demons or magic would turn me into a cultist.

Special mention for the N64 as well. I have countless memories of playing 4-player games with friends, primarily Goldeneye (though Perfect Dark was better and I will die on that hill).