Xbox Live is an online gaming service that works across both the current Xbox system and the future Xbox 2. You will be able to play online and compete against others across both consoles. If you are playing an Xbox game on Live you will be able to compete against people playing that same game on Xbox 2
Xbox 360 is Backwards Compatible
Source: The Inquirer
It looks like the new Xbox 360 will be able to play the last generations Xbox games, a trend started by the PS2 console. I can’t imagine how this would be really difficult though as the Xbox consoles are so close to a PC in their technology.