FaceTime is Apple's video calling service. It allows anyone with a recent iPhone, iPod touch, iPad, or Mac to make free calls to any other Apple user over Wi-Fi or cellular connection. On the iPhone, FaceTime is bundled into the Phone app. On the iPod touch, iPad, and Mac, FaceTime is a separate app. On all devices, FaceTime can connect to any Apple-registered iPhone phone number or email address. That makes it perfect for keeping in touch with family who lives far away, with the kids while traveling, with business partners at distant offices, or even with that special someone while shopping for the perfect gift.
With iOS 7, Apple added FaceTime Audio as a way to make voice calls over the same network.
Apple originally announced that they would release FaceTime as an open standard so that third parties to create FaceTime for Android, FaceTime for PC, etc. So far, that hasn't happened. Apple was sued by a patent troll of FaceTime and, after losing on appeal, Apple has been re-architecting FaceTime to work through relay servers. (This could be why FaceTime was never released — it would be subject to the same patent trolling on any other platform.)