This is the never-ending discussion.
To be honest both work in the same manner, each with their advantages and disadvantages:
- If you want a console for pure online gaming, or mostly online gaming, Xbox. However, the downside of it is that you have to pay for a membership yearly. The PS3 has got free online but as it's free it can be more buggy (I haven't had a problem as of yet with it but I know people have had issues).
- If you want a console for offline games like Elder Scrolls, Final Fantasy, etc, go for the PS3. This might be a bit biased, because essentialy, both consoles work in the same manner and the same game in both consoles looks 99.9999% the same. I just recommend you get one of the better built Xbox consoles like an Elite, because some games need more memory (games with big cities and a lot of people, trees, etc). The normal 360 tends to get loading screens whereas the PS3 doesn't. Though I'm not sure what happens on the Elite, but I'd think it's the same as with the PS3.
- If you want an entertainment console in which you can watch films and play a game or two and maybe go on the web, go for the PS3. Bluray films look amazing, given you own a plasma TV. Getting the PS3 also saves you from buying a Bluray player on top of the console. Also, you can use your PS3 literally as a computer and browse the web. There's also a mod for it which installs a different OS (never tried it).
One thing you must look out for though is the type of games you want to play. Each console has exclusive games as well as the multiplatform ones. So make a list of the games you'd like to play and see which console can play most of them.
Just as an example: The Xbox has Halo, Forza Motorsport, Left 4 Dead. The PS3 has Gran Turismo, God Of War, Metal Gear.
That's all I can say really