The feature that lets you start a task on one Apple device and continue it on another is Handoff. It lets a person move an activity—such as browsing a webpage, writing an email, or editing a document—from one nearby Apple device to another and resume where they stopped.
Handoff is part of Continuity and works with Apple apps including Safari, Mail, Maps, Notes, Calendar, Pages, Numbers, and Keynote, as well as participating third-party apps. The devices must normally be compatible, signed in to the same Apple Account, and have Wi-Fi, Bluetooth, and Handoff enabled. The receiving device displays a Handoff icon, such as in the Mac Dock or iPhone app switcher.
Handoff is often confused with Universal Clipboard because both connect Apple devices. Universal Clipboard moves copied content; Handoff moves an in-progress app activity. Sidecar is different again: it uses an iPad as an additional display for a Mac. Continuity is the broader family of Apple features that includes Handoff, Universal Clipboard, Sidecar, and several other cross-device capabilities. Handoff debuted with Apple’s OS X Yosemite and iOS 8 generation in 2014.