Sadly, cross-platform play will apparently not be a feature.
If you want to play Xbox 360 games on your PC at the moment, then you have to jump through all kinds of hoops involving emulators and such, and that's not exactly an ideal solution. But according to insider sources, when the new version of Windows arrives, all that will change.
According to the rumors, Windows 8 will be able to run Xbox 360 games without the need to mess around with emulators. There is the suggestion that anyone actually wanting to make use of this feature would have to pay a subscription fee, although it's possible that that just refers to playing online, as is the case with Xbox Live. People playing on a PC apparently won't be able to play with their console brethren either, as multiplayer will use the Windows Live service, rather than Xbox Live.
If this rumor turns out to be true, it would be an interesting move on Microsoft's part. Hundreds of millions of people use Windows, and while obviously not every Windows 8 user will want to play Xbox 360 games - or indeed, be able to - it has the potential to massively increase the install base for 360 games. Interestingly, Microsoft has recently folded its Games for Windows Live site into Xbox.com, which might mean nothing, or it might be a sign that this rumor has something to it.
Microsoft has yet to confirm when Windows 8 is coming out, but most rumors point to an October 2012 release.