The idea of continuity between devices has been around for some time now and is improved upon with each new iOS iteration. Basically it allows you to quickly and seamlessly transfer your work, calls, messages, etc. from your iPhone or iPad to your Mac or vice versa. One particularly useful feature on both the Safari and Chrome browsers lets you transfer your tabs, so if you've navigated to a webpage on your phone, you can simply open up the same page on your Mac without having to do something tedious like copy the link and email it to yourself, as you may have done once upon a time.
Along with the addition of Siri and Apple Pay for Safari, macOS Sierra introduces the Universal Clipboard, which allows you to share your copy and paste clipboard between your devices. This is another Continuity feature like Handoff that allows you to seamlessly start a task on one device and continue it on another.
Handoff is one of the features of the Continuity suite that debuted with iOS 8 that lets you pass information between apps on different devices using a Bluetooth connection. For example, you could start an email on your iPhone, then hand it off to your Mac and continue from there. Handoff works with a number of apps including Mail, Messages, Safari, Maps, Notes, Calendar, Contacts, Keynote, Pages, Numbers, Phone and Reminders, as well as some third party apps.
Handoff is a feature in iOS 8.0 that allows you to start a document or email on your iPhone and complete it on your Mac, iPad or iPod touch. Handoff is only supported on these devices:
Handoff is a part of "Continuity" introduced by Apple with iOS 8 and OS X Yosemite. Continuity allows users to "seamlessly" move between their iOS devices and Mac, according to Apple. For example, iPhone users can receive and make calls on their iPad or iPod touch using Continuity.