Well, if we're talking about current gen consoles, I'm going to have to say the PS3. The 360 is a great console, don't get me wrong, and it also brought some things into play that became a staple, like Achievements, which the PS3 later implemented in the form of Trophies, and its online play with XBOX Live. While some will say the Dreamcast introduced online console gaming, it was Microsoft that got it right with gamer accounts, marketplace, etc..
So why do I like the PS3 more? Simply because of its familiarity with the PS2 and PS1 as far as controller feel go. Its exclusives are great, whether it's God of War, Uncharted, Ratchet & Clank, LittleBigPlanet, etc., it has its share of great exclusives. The PSN's online gaming is free, which is a plus, though it isn't as good as the XBOX's, so it does come with a sacrifice. While not the most important aspect of a game, the PS3's graphical power is a beast, but only shows this with exclusives made for the PS3 only, while multi-platforms, depending on the games, perform better on the XBOX (Bayonetta and Skyrim are perfect examples). Overall, it boils down to personal preference, and I happen to prefer the PS3 over the 360. Does that mean the PS3 is better? Absolutely not, as "better" or "worse" are completely subjective in this case.
As for my favorite system spanning all generations, it's a very tough call, but I have to say it's the Super Nintendo, with the Playstation coming in at a very close (and I mean VERY close) second. They both had a legendary library of games spanning nearly every genre, so there really was something for everyone. The SNES had games like Chrono Trigger, Mario Kart, Super Metroid, and Megaman X (just to name very few), while the Playstation gave way to Final Fantasy VII and onward, Metal Gear Solid, Twisted Metal, and Resident Evil, among so many other games. Along with the impact they had on the industry, whether it was the SNES's Mode 7 graphics and 8 channels of sounds, or the Playstation's disc-based games and capability for some amazing CG FMVs, they also have games that I can pop in and play all over again and never get bored. Those two systems really can stand the test of time, and they have certainly aged rather well.