I've had the hand-held "consoles" such as a Sega Mega Drive and Game Boy, but I wouldn't necessarily class them as consoles.
Apart from them, I've owned a PS1, PS2 and now an Xbox 360. Not a huge selection really so I can comfortably say I've loved them all. I grew up on the PS1 and it was the birth of many great series of games such as Gran Turismo, and several wrestling games like WCW Nitro. Tony Hawks Skateboarding was born as well as Track & Field being released which was quality with 4+ mates in one room tapping away trying to hold back laughter.
Whereas the PS2 give us the likes of Smackdown 2 (I think) which was excellent, nothing better than giving someone The Rockbottom in a kitchen backstage. Gran Turismo 3 was released as well, quite easily the greatest driving game ever made. GTA San Andreas was also released on the PS2, which is the most enjoyable GTA that I've played. Not only that, there was many superb PES titles on the PS2, when the PES series was in it's prime no doubt. A bonus was that the PS2 was so easy to "chip" or "modify", I didn't have mine chipped but I did get it modified which meant I could play patched PES games, especially the Winning Eleven series from Japan.
Xbox 360 = Ultimate online experience. Playing the likes of COD4 online is truly enthralling, especially with plenty of friends, it's always absolutely hilarious. BioShock is also worthy of a mention because it's one of my favourite single player experiences ever. PES6 was also superb online, shit single player but online was so much fun, I must have played hundreds of matches and was even in an online PES League, before Konami ruined the online with PES08 and 09. Skate also was a very refreshing sports game to play using the analog sticks revolutionarily.
I've probably missed quite a few classics out but, I only have so much time to waste. As for my favourite console, it's probably between the PS2 and Xbox 360.