XBox vs PS2: XBox was faster, but PS2 launched earlier, was cheaper, and had more games, so won the generation.
XBox 360 vs PS3: Sony reacts to Microsoft by making the PS3 more powerful than the 360 and more expensive. Microsoft reacts by getting more games and really doubling down on Halo (the biggest draw to XBox back then). XBox wins the generation.
Note: this is from memory, didn't check sources!