In Short
Continuum is one of Microsoft's signature features in Windows 10 where the user interface can adapt automatically depending on the situation. For example, on a Surface Pro 3, Windows 10 will default to desktop mode with the keyboard attached and will then switch to tablet mode when you remove it.
Continuum will also be present in Windows 10 Mobile allowing your smartphone to attach to a monitor, mouse and keyboard, wired through the use of accessories or wireless through Bluetooth and Miracast. It will then project a desktop like app experience to the monitor using the apps on your phone.
Here's how Microsoft describes Continuum:
"Windows 10 adjusts your experience for your activity, device and display, so you can do your thing in any mode anytime you want. Onscreen features, like menus and task bars, adapt for easy navigation. Apps are built to scale smoothly from screen to screen so they look good from the smallest app window up to the largest 8k displays. You can even change from desktop to tablet mode anytime you want – your screen will give you a smooth transition and a beautiful display. Available exclusively on Windows, Continuum is designed to ensure that the best screen is always the one you're on."
Continuum for Windows 10 is available to use now and will come to market with Windows 10 Mobile later in 2015. On phones, however, it will require new hardware, so existing phones won't support it. It requires the power to drive dual displays, something not present in current Windows Phones.
