I forgot to mention the PSP. I'd think I'd tack on the WonderSwan Color, Sega Saturn and Playstation 2 too (PS2 has backwards compatibility with the PS1).
I've noticed I'm the only one who mentioned the not as popular systems like the Sega Game Gear and WonderSwan. I didn't own either, but from what I've played through emulation, many of the games on them are nice.
The Game Gear has an awesome library, but holy shit was it a terrible system. Six AA batteries gone in an hour, it weighs about four pounds, it's almost the size of the Wii U gamepad with none of the comfortable finger holds. I love the games, and its audio/video capabilities were unrivaled at the time, but the technology just wasn't ready for the ambition.
The Wii itself had a pretty disappointing library imo, but the fact that it's backwards compatible with GCN and has Virtual Console puts it near the top of my list. I mean It's so amazing to have every console Zelda game in one place thanks to things like USBLoader~
(Apparently the Wii U can do all that too and I love everything they've released for it so far, thus I put it at #1)
The PS2 is really high up there too for having so many great games (Okami, Shadow of The Colossus,etc.) and full PS1 compatibility. I also really love the Dualshock controller.
GBA/DS are the best handhelds, GBA because it has a great library of it's own and plays GB/GBC. DS because it has an absolutely massive library, makes good use of the dual screens and plays GBA games of course.