It's a way to connect your Windows Mobile phone with your Xbox Live account, to an extent:
Microsoft will also roll out a free Xbox Companion app for Windows Phone that will let users discover, learn about, and control content from the new entertainment services on Xbox Live.
MajorNelson posted an article about it as well. The companion app seems to be able to access and control a bunch of features on the Xbox, like sort of a terribly overpriced Xbox remote control. For instance, you can navigate the dashboard, or fast forward/rewind a video that's playing, etc.
Since this might not be what you want all the time, there's an option in the preferences to disable the Companion app's ability to control the console.
In addition to Windows Phone support, there's a more stripped down version for iOS devices. I don't think the "Xbox Companion" functionality is really there, but you can at least mess with your profile and look at achievements.