I struggle with this all the time as I'm never 100% confident in my choices, but I'll say in no specific order...
Playstation 2 - Great selection of games from racing, to action, to JRPG, to the start of the Guitar Hero era of gaming which was real big for me.
Nintendo 64 - I'll never say this is an "underrated" system or it was wrongly criticized, the controller sucks and it has a very small selection of games, but this system has all time favorite games for me like Banjo-Kazooie, Ocarina of Time, Super Mario 64, Diddy Kong Racing, Goldeneye, Perfect Dark, etc...
Xbox 360 - This was that big leap to the HD era for me, which I think has a lot of importance to it. I was a big PS2 player before this, so moving from Sony to Xbox fully was a big shift and there were lots of great games I played, continued love for Rock Band and Guitar Hero, and this was a bigger push into online console gaming as I was primarily a PC online player outside of some online stuff I did for like SOCOM in the PS2 days or the bit of time I put into Final Fantasy XI. Also it has one of the top controllers ever made. This is also one of the only original consoles I own still (Not the original original one, I had to replace it with a new one in like 2009 or 2010 after something died on it and I had to buy a new Arcade version to put my hard drive into), as I like to plug it back in and play Guitar Hero and Rock Band usually once or twice a year.
NES - This one is more for pure nostalgia than say the Nintendo 64, but it's an important system for me, having the first games I've ever played, having important games for my gaming interests like Super Mario Bros 1/3, Legend of Zelda, and while there aren't tons of games I could point to as still being my favorites or having continued importance, it always has a soft spot for me.
Honorable Mention:
Playstation 1 - This one actually has a lot of importance to me, but it was a system I never owned, I played it at friends houses, a small card shop in my town, and I got super into games like Twisted Metal 2, Crash Bandicoot, Tekken 3, Tony Hawk Pro Skater, Final Fantasy 7 and lots more that were pretty significant games for me, offering different experiences than stuff I had with N64.