There are three main ways to invite people into a Google Hangout and Google Hangout on Air: e-mail invite, invite via link, or invite within the Hangout. I will be talking about the third one. There are some people that only have a phone or a tablet and doing the other two ways doesn’t really work in my experience. But the third way works!
It’s easy to do when you are in a Hangout and can be done by anyone in it, not just the host. There is a “add person” button where you can (un)mute the mic, turn on or off the cam, ect as in the screenshot below: