A window pops up that lets you choose exactly which window you want to share. One of the options is Desktop, which includes everything you can see on your screen. Choose either Desktop or a specific window you want to show people, and then click the "Share selected window" button. The window you picked appears in place of your video feed.
If you pick Desktop, things can get a bit tricky, since the folks in the hangout might end up seeing a duplicate of the hangout window, and you have to be conscious of everything you're showing these folks and keep in mind that they can only see a smaller, less clear version of your screen. So it's usually better to share a single window, and use windows and apps with big fonts and large buttons. You can play a video on your shared screen, but it might look really choppy and blurry to the other people in the hangout.
Hangouts on Mobile Devices
If your Android phone or tablet has a front-facing camera (that is, it can take photos of you when you're looking at its screen), and you're running a relatively modern version of the Android operating system, you can take part in Hangouts right through your device -- no keyboard or monitor required.
To get started, launch the Google+ mobile app. Then simply click the "Join hangout" button in a stream you're checking out, or (if you receive a notification about a hangout) tap the notification. You see the same "Check your hair" screen you get on a computer. Tap the "Hang out" button to join the fun.
The hangout's main video feed (the one showing the person who's talking) pretty much takes over your device's entire screen. You can check how you look by glancing at the small box in the lower-right corner. The buttons at the bottom let you read and or send chat messages, mute your mic or video, switch to your device's back-side camera, and exit the hangout.
Tips for Better Hangouts
If you're going to spare the time -- and the computer power -- to set up a hangout with a group of friends and acquaintances (and, if you're feeling brave, complete strangers via public hangouts), you may as well do it right. That means having the best possible connection, decent lighting, and the right spirit and etiquette to converse and lead the group along. Some hangouts are meant to be casual and loosely structured, but every hangout can benefit from a little know-how.
Here are a few tips to help you get the most out of your group video chat, drawn from your humble author's experience on live-via-Skype talk shows.
Dig out an Ethernet cable. A wireless connection is okay for most video-chat situations. But to avoid choppiness and get the absolute smoothest video, grab a network cable and connect your laptop or desktop computer directly to one of the outgoing ports on your home router. Video connections suffer a bit from the back-and-forth bounce of a WiFi signal, but a cable connection should give you good video quality as long as you have a fast Internet connection (see the Note on Note for the minimum requirements).
Grab a USB headset. You'll get the best possible sound if you use a head-set that plugs into a USB port on your computer. The next best thing is to use standard earbud or in-ear headphones (this can help prevent echoes and feedback). If you don't have either of those, go ahead and talk straight at your laptop, but at least determine where your computer's microphone is so you know where to aim. (Finding the mic can be tricky -- try Googling your computer's model number or flip through its manual)
Speak in full sentences, and wait an extra second before talking. If you've ever been on a conference call on a glitchy day or made a really long-distance international phone call, you've experience the frustration of talking over the end of someone's sentences, and then apologizing, and then apologizing again for talking over their apology, and so on. That can occasionally happen in hangouts, especially if more than two people have joined. To help avoid that problem, try to speak in full, thought-out sentences with clear conclusions, and occasionally remind others to try the same.
Get lots of light behind your camera. Webcams don't do so hot in low lighting, and your face can end up indistinct if the light is behind your head. Try to set things up so there's an ample light source behind the camera, leaving your face illuminated but not washed out.
The best way to get better at hangouts is to give them a try. Talk to some friends, meet some friends-of-friends, and get used to the idea that you can use Google+'s video-chat service in a way that doesn't feel like an job interview. Line up the camera and get to know the people in your circles.