Across the top of my copy of Mass Effect 3 is a purple "Works better with Kinect" banner. This seems relatively out of place in a 3rd person shooter. However, I do own a Kinect sensor, and I'm tempted to hook it up.
What can I do with Kinect that I couldn't do otherwise?
Hooking up the Kinect allows you to issue voice commands to your squad during fighting, as well as complete actions such as opening doors, examining objects, etc. In addition, you can apparently choose the dialogue by speaking the shortcut phrase. Lacking a Kinect to test this, I can't verify how well it responds.
Here is a cheatsheet list of commands (with typos!):
Basically, it lets you issue squad commands without having to bring up the power wheel in order to direct them, and make conversation choices without having to click.
You are able to issue squad combat commands and inventory commands.
Here is a full commands guide: http://www.gamefront.com/mass-effect-3-kinect-commands-guide/
It is discussed in depth here:
It's only for voice commands. For instance, you can open doors or lockers, select dialog options, switch weapons, and issue commands to your squad.
