I got the original XBox to play Shenmue in English. I'd played it in JP on the Dreamcast. When the PS3 & 360 came out, I played the PS3 almost exclusively. I didn't have a NA XBox 360 only a JP version. I used it to play SHMUPs because the JP 360 was the only console that catered to that genre. When a NA special FF XIII white 360 console came out, I bought it. After playing it a while, I felt that although the PS3 was the technically superior console that the 360 had a better user interface and community experience. Beginning in 2011, I spent 90% of my time on the 360 and only about 10% on the PS3.
I thought that after all of us owning the PS2, we'd all go PS3. I got it day one, and they all went 360. It was a while before I got one myself though.
A mate at work was talking up Lost Planet, and after a few weeks of his mutterings I thought "why not" and got an Elite with Blue Dragon and bought LP along side it. I also think around this time was leading up to the first Mass Effect, which made the decision much easier - I knew I wanted that game as soon as I saw it.
It wasn't till the very end of '07 did I heavily invest in the 360 though, with friends going on and on about Call of Duty 4: Modern Warfare. After being one of those who didn't like the idea of online consoles, I didn't realise how addictive it could be. One thing I am glad to be wrong about. :P
My younger brother had the original Xbox (got that 'Limited Edition' crystal variant), got the usual suspects; Halo 1 and 2, Forza Motorsport, Spinter Cell, as well as San Andreas and The Godfather. Didn't enjoy the Halo's or Splinter Cell. Only played SA because of being able to listen to your own music. I'd say Forza and God Father were my favourite games on it, but not enough to make me want one.