Messages: The OS X version of the Messages app can use Apple's iMessage protocol for text and MMS messages, but for video calls or screen sharing, you must use an account type that supports video (namely, AIM, Jabber, or Google Talk). You can get free accounts for any or all of these services, and set them up in Messages > Preferences > Accounts.
Before starting a video call, you must put the other person on your buddy list — but you can add them unilaterally. Although you can have video calls with up to three other people, screen sharing is possible only when you're on a call with one other person. Note, also, that the iOS version of Messages does not support video calls; to use a service like Google Talk on your iOS device, you'll need a third-party app such as the free Vtok.
FaceTime: FaceTime, on either OS X or iOS, is great for one-on-one audio or video calls. Because it's simple to use and available almost anywhere, it's an ideal choice if you know the other person is an Apple user. And it offers highly secure end-to-end encryption. In most cases, you can use either an email address or telephone number to initiate a call, and the other party need not have FaceTime open or do anything special to log in. But you can't have more than two participants in a video call, and screen sharing isn't available.
Skype: With support for many platforms, multi-person video, and screen sharing (with simultaneous video), Skype is a great all-purpose choice for video calls, and it offers encryption (although with fewer protections than FaceTime). But it comes with a few gotchas. Before you can call someone, you may need their approval to add them as a contact (depending on their privacy settings). That's fine for friends and business colleagues, but if you're calling someone who doesn't recognize your name, there's no guarantee they'll accept you as a contact. Furthermore, the other party must be logged in to Skype on at least one device.
Google+ Hangouts: Like Skype, Google+ Hangouts can be used on a variety of platforms. On OS X, you log in to your Google account in a Web browser. All participants need a Google account as well as the Google Voice and Video Plugin (which you're prompted to install the first time you start or join a video call). iOS users need the free Hangouts app. Using Google+ Hangouts you can share your Mac's screen and have video calls with up to nine other people. If any of the other participants aren't signed in to Hangouts on at least one device, they'll receive a notification when you try to call them (which they may or may not see immediately).
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