All AV cables designed for the Xbox 360 are compatible with all systems. Standard HDMI cables will require an HDMI port though which is something that early Xbox 360's lacked.

All power supplies aren't compatible with all systems. They've been reducing the power needs of the console periodically. You can use a higher rated power supply from an older console on a more recent console with lower power needs since it will just take what it needs.

But you can't use a lower rated power supply from a more recent system on a older console with higher power requirements. The power output wouldn't be adequate and furthermore, they've designed the power plugs to make it physically impossible to ever plug an incompatible 360 power supply into a console it can't support.

Not enough detail in your post to give you an answer. Check the power requirements on your console and make sure his is rated for as high or higher. And make sure if the AV cable you're recieving is HDMI that your console has an HDMI port.