My favorite game console would have to be the SNES, but the NES, which is where I started, is not far behind. I'm also really fond of the PlayStation 1, which I consider to be the rightful successor to the SNES, the Sega Genesis, which really gave the SNES a run for its money, and the Sega Dreamcast, which was criminally underrated and got more good games in a shorter period of time than any other console I've ever seen.
My favorite handheld is the PSP, which I didn't expect I would like, but completely blew me away. As much as I love the original GameBoy's library, I'll take my GBA SP any day. I'd also like to say that the Neo Geo Pocket Color and the Atari Lynx are both really good machines that were largely ignored.
My favorite game is a toss up between Secret of Mana for the SNES and Castlevania: Symphony of the Night for the PS1. Other games I really like include Centipede, Thunder Force V, the original Legend of Zelda, Breath of Fire, Quake II, Seiken Densetsu 3, Mushihimesama Futari, anything by Yasumi Matsuno, etc.
Favorites by console (tip of the iceberg):
Atari 2600: Demons to Diamonds, Turmoil, Galaxian
Jaguar: Tempest 2000
Colecovision: Frenzy, Pepper II, Mr. Do
NES: Legend of Zelda, Mega Man 3, Zelda II, Contra, Adventure Island II, Tiger Heli, SMB, Faxanadu, too many to name...
SNES: Secret of Mana, Super Metroid, Tetris Attack, Breath of Fire, Mega Man X, Super Mario World, R-Type III, moar
N64: Beetle Adventure Racing, Robotron 64, Super Smash Bros.
Gamecube: F-Zero GX, Super Smash Bros. Melee, Viewtiful Joe, Wind Waker, Phantasy Star Online, Metroid Prime
Master System: Golvelius, Power Strike, Phantasy Star
Genesis: Toejam & Earl, Thunder Force III, LandStalker, Shinobi III, Sonic the Hedgehog, Streets of Rage 2, Fire Shark
Sega CD: Lords of Thunder, Popful Mail, Final Fight CD
32X: Kolibri
Saturn: Thunder Force V, Shining the Holy Ark, Shienryu, Panzer Dragoon Zwei, Necronomicon, Galactic Attack
Dreamcast: Bangai-O, King of Fighters: Dream Match 1999, ChuChu Rocket, Skies of Arcadia, SoulCalibur
PlayStation: Castlevania: SoTN, Final Fantasy IX, SoulBlade, Wipe'Out", Vagrant Story, Spyro the Dragon, Einhänder, etc.
PlayStation 2: Culdcept, R-Type Final, Dragon Quest VIII, Metal Slug Anthology, Atelier Iris, Ōkami
PlayStation 3: 3D Dot Game Heroes, Demon's Souls, Dark Souls, Mortal Kombat, Outland
Xbox: Halo, Morrowind, PGR2
Xbox 360: Raiden Fighters Aces, Mushihimesama Futari, PGR4, Borderlands, Dead Space, Oblivion, Pinball Hall of Fame
Lynx: Rygar
GameBoy: Centipede/Millipede
Game Gear: Sonic the Hedgehog
Neo Geo Pocket Color: Match of the Millenium
GameBoy Advance: Yggdra Union
PSP: Gradius Portable
DS: Etrian Odyssey
I think the point is I like video games.