With Cortana, you can use a headset with mic for voice control in addition to a Kinect sensor. It works the same for both, with one exception: “Hey Cortana, Xbox on” only works with a Kinect. So you’ll have to turn on your Xbox the old fashioned way—with a controller—if you’re using a headset.
Just about any headset with a 3.5 mm headphone jack will work. Certain controllers may need an Xbox One Stereo Headset Adapter to attach their headset. See Connecting a compatible headset to Xbox One at Xbox.com for more info.
If you have two or more headsets connected at the same time and aren’t sure which one has voice control, look at the profile signed in at the top-left corner of the Home screen. The person signed in is the one with voice control.